Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Fundamentals of Biostatistics
The data in Table 2.13 are a sample from a larger data set collected on people discharged from a selected Pennsylvania hospital as part of a retrospective chart review of antibiotic usage in hospitals [7]. The data are also given in Data Set HOSPITAL.DAT with documentation in HOSPITAL. DOC at www.cengagebrain.com. Each data set at www.cengagebrain.com is available in six formats: ASCII, MINITAB-readable format, Excel-readable format, SAS-readable format, SPSS-readable format, and Stata-readable format, and as a text file (R-readable format). Compute the mean and median for the duration of hospitalization for the 25 patients.The data in Table 2.13 are a sample from a larger data set collected on people discharged from a selected Pennsylvania hospital as part of a retrospective chart review of antibiotic usage in hospitals [7]. The data are also given in Data Set HOSPITAL.DAT with documentation in HOSPITAL. DOC at www.cengagebrain.com. Each data set at www.cengagebrain.com is available in six formats: ASCII, MINITAB-readable format, Excel-readable format, SAS-readable format, SPSS-readable format, and Stata-readable format, and as a text file (R-readable format). Compute the standard deviation and range for the duration of hospitalization for the 25 patients.It is of clinical interest to know if the duration of hospitalization is affected by whether a patient has received antibiotics. Answer this question descriptively using either numeric or graphic methods. Infectious Disease The data in Table 2.13 are a sample from a larger data set collected on people discharged from a selected Pennsylvania hospital as part of a retrospective chart review of antibiotic usage in hospitals [7]. The data are also given in Data Set HOSPITAL.DAT with documentation in HOSPITAL. DOC at www.cengagebrain.com. Each data set at www.cengagebrain.com is available in six formats: ASCII, MINITAB-readable format, Excel-readable format, SAS-readable format, SPSS-readable format, and Stata-readable format, and as a text file (R-readable format). TABLE 2.13 Hospital-stay dataSuppose the scale for a data set is changed by multiplying each observation by a positive constant. What is the effect on the median?Suppose the scale for a data set is changed by multiplying each observation by a positive constant. What is the effect on the mode?Suppose the scale for a data set is changed by multiplying each observation by a positive constant. What is the effect on the geometric mean?Suppose the scale for a data set is changed by multiplying each observation by a positive constant. What is the effect on the range?A man runs 1 mile approximately once per weekend. He records his time over an 18-week period. The individual times and summary statistics are given in Table 2.14. What is the mean 1 mile running time over 18 weeks? Table 2.14 One mile running time for an individual, over 18 weeksA man runs 1 mile approximately once per weekend. He records his time over an 18-week period. The individual times and summary statistics are given in Table 2.14. What is standard deviation of the 1 mile running time over 18 weeks? Table 2.14 One mile running time for an individual, over 18 weeksA man runs 1 mile approximately once per weekend. He records his time over an 18-week period. The individual times and summary statistics are given in Table 2.14. Table 2.14 One mile running time for an individual, over 18 weeks Suppose we construct a new variable called time_100 = 100 time (e.g., for week 1, time_100 = 1280). What is the mean and standard deviation of time_100?A man runs 1 mile approximately once per weekend. He records his time over an 18-week period. The individual times and summary statistics are given in Table 2.14. Table 2.14 One mile running time for an individual, over 18 weeks Suppose we construct a new variable called time_100 = 100 time (e.g., for week 1, time_100 = 1280). Construct a stem and leaf plot of time_100 using the first 3 most significant digits for the stem and the least significant digit for the leaf. So, for week 1, time_100 = 1280 which has a stem = 128 and a leaf = 0.A man runs 1 mile approximately once per weekend. He records his time over an 18-week period. The individual times and summary statistics are given in Table 2.14. Table 2.14 One mile running time for an individual, over 18 weeks Suppose we construct a new variable called time_100 = 100 time (e.g., for week 1, time_100 = 1280). Suppose the man does not run for 6 months over the winter due to snow on the ground. He resumes running once a week in the spring and records a running time = 12.97 minutes in his first week of running in the spring. Is this an outlying value relative to the distribution of running times recorded the previous year in Table 2.14? Why or why not? Hint: Construct a Box plot based on the data in Table 2.14, and assess whether this new point is an outlier based on Definition 2.11.The data in Table 2.15 are a sample of cholesterol levels taken from 24 hospital employees who were on a standard American diet and who agreed to adopt a vegetarian diet for 1 month. Serum-cholesterol measurements were made before adopting the diet and 1 month after. The data are available at cholesterol.xls at www.cengagebrain.com. Table 2.15 Serum-cholesterol levels (mg/dL) before and after adopting a vegetarian diet Compute the mean change in cholesterol.The data in Table 2.15 are a sample of cholesterol levels taken from 24 hospital employees who were on a standard American diet and who agreed to adopt a vegetarian diet for 1 month. Serum-cholesterol measurements were made before adopting the diet and 1 month after. The data are available at cholesterol.xls at www.cengagebrain.com. Table 2.15 Serum-cholesterol levels (mg/dL) before and after adopting a vegetarian diet Compute the standard deviation of the change in cholesterol levels.The data in Table 2.15 are a sample of cholesterol levels taken from 24 hospital employees who were on a standard American diet and who agreed to adopt a vegetarian diet for 1 month. Serum-cholesterol measurements were made before adopting the diet and 1 month after. The data are available at cholesterol.xls at www.cengagebrain.com. Table 2.15 Serum-cholesterol levels (mg/dL) before and after adopting a vegetarian diet Construct a stem-and-leaf plot of the cholesterol changes.The data in Table 2.15 are a sample of cholesterol levels taken from 24 hospital employees who were on a standard American diet and who agreed to adopt a vegetarian diet for 1 month. Serum-cholesterol measurements were made before adopting the diet and 1 month after. The data are available at cholesterol.xls at www.cengagebrain.com. Table 2.15 Serum-cholesterol levels (mg/dL) before and after adopting a vegetarian diet Compute the median change in cholesterol.The data in Table 2.15 are a sample of cholesterol level: taken from 24 hospital employees who were on a standard American diet and who agreed to adopt a vegetarian die for 1 month. Semm-cholesterol measurements were mad before adopting the diet and 1 month after. The data art j. available at cholesterol-xls at www.cengagebrain.com. TABLE 2.15 Serum-cholesterol levels (mg/dL) before and after adopting a vegetarian diet Some investigators believe that the effects of diet on cholesterol are more evident in people with high rather than low cholesterol levels. If you split the data in Table 2.15 according to whether baseline cholesterol is above or below the median, can you comment descriptively on this issue?In an experiment that examined the effect of body position on blood pressure [8], 32 participants had their blood pressures measured while lying down with their arms at their sides and again standing with their arms supported at heart level. The data are given in Table 2.16. They are also available at Table 2.16 Effect of position on blood pressure Compute the arithmetic mean and median for the difference in systolic and diastolic blood pressure, respectively, taken in different positions (recumbent minus standing).In an experiment that examined the effect of body position on blood pressure [8], 32 participants had their blood pressures measured while lying down with their arms at their sides and again standing with their arms supported at heart level. The data are given in Table 2.16. They are also available at www.cengagebrain.com. Orthostatic hypertension is sometimes defined based on an unusual change in blood pressure after changing position. Suppose we define a normal range for change in systolic blood pressure (SBP) based on change in SBP from the recumbent to the standing position in Table 2.16 that is between the upper and lower decile. What should the normal range be?Table 2.17 Format for FEV.DAT The data in Table 2.17 are available for each child. For each variable (other than ID), obtain appropriate descriptive statistics (both numeric and graphic).Forced expiratory volume (FEV) is an index of pulmonary function that measures the volume of air expelled after 1 second of constant effort. Dataset FEV.DAT at www.cengagebrain.com. contains determinations of FEV in 1980 on 654 children ages 3 through 19 who were seen in the Childhood Respiratory Disease (CRD) Study in East Boston, Massachusetts. These data are part of a longitudinal study to follow the change in pulmonary function over time in children [9], TABLE 2.17 Format for FEV.DAT Column Variable Format or code 1-5ID number 7-8Age (years) 10-15FEV (liters) X.XXX 17-20Height (inches) XX.X 22 Sex0 = female/1 = male 24 Smoking status0 = noncurrent smoker/ 1 = current smoker The data in Table 2.17 are available for each child. Use both numeric and graphic measures to assess the relationship of FEV to age, height, and smoking status. (Do this separately for boys and girls.) Use both numeric and graphic measures to assess p the relationship of FEV to age, height, and smoking status. (Do this separately for boys and girls.)Forced expiratory volume (FEV) is an index of pulmonary function that measures the volume of air expelled after 1 second of constant effort. Dataset FEV.DAT at www.cengagebrain.com. contains determinations of FEV in 1980 on 654 children ages 3 through 19 who were seen in the Childhood Respiratory Disease (CRD) Study in East Boston, Massachusetts. These data are part of a longitudinal study to follow the change in pulmonary function over time in children [9], Compare the pattern of growth of FEV by age for boys and girls. Are there any similarities? Any differences? Hint: Compute the mean FEV by age group (3-4/5-9/10- 14/ 15-19) separately for boys and girls and plot the meanFEV by age.26PThe food-frequency questionnaire (FFQ) is an instrument often used in dietary epidemiology to assess consumption of specific foods. A person is asked to write down the number of servings per day typically eaten in the past year of over 100 individual food items. A food-composition table is then used to compute nutrient intakes (protein, fat, etc.) based on aggregating responses for individual foods. The FFQ is inexpensive to administer but is considered less accurate than the diet record (DR) (the gold standard of dietary epidemiology). For the DR, a participant writes down the amount of each specific food eaten over the past week in a food diary and a nutritionist using a special computer program computes nutrient intakes from the food diaries. This is a much more expensive method of dietary recording. To validate the FFQ, 1 73 nurses participating in the Nurses Health Study completed 4 weeks to diet recording about equally spaced over a 12-month period and an FFQ at the end of diet recording [10]. Data are presented in data set VALID.DAT at www.cengagebrain.com for saturated fat, total fat, total alcohol consumption, and total caloric intake for both the DR and FFQ. For the DR, average nutrient in- takes were computed over the 4 weeks of diet recording. Table 2.18 shows the format of this file. Use descriptive statistics to relate nutrient intake for the DR and FFQ. Do you think the FFQ is a reasonably accurate approximation to the DR? Why or why not?In Section 2.9, we described Data Set LEAD.DAT (at www .cengagebrain.com) concerning the effect of lead exposure on neurological and psychological function in children. Compare the exposed and control groups regarding age and gender, using appropriate numeric and graphic I descriptive measures.In Section 2.9, we described Data Set LEAD.DAT (at www.cengagebrain.com) concerning the effect of lead exposure on neurological and psychological function in children. Compare the exposed and control groups regarding verbal and performance IQ, using appropriate numeric and graphic descriptive measures.Activated-protein-C (APC) resistance is a serum marker that has been associated with thrombosis (the formation of blood clots often leading to heart attacks) among adults. A study assessed this risk factor among adolescents. To assess the reproducibility of the assay, a split-sample technique was used in which a blood sample was provided by 10 people; each sample was split into two aliquots (sub-samples), and each aliquot was assessed separately. Table 2.19 gives the results. Table 2.19 APC resistance split-samples data To assess the variability of the assay, the investigators need to compute the coefficient of variation. Compute the coefficient of variation (CV) for each subject by obtaining the mean and standard deviation over the 2 replicates for each subject.Activated-protein-C (APC) resistance is a serum marker that has been associated with thrombosis (the formation of blood clots often leading to heart attacks) among adults. A study assessed this risk factor among adolescents. To assess the reproducibility of the assay, a split-sample technique was used in which a blood sample was provided by 10 people; each sample was split into two aliquots (sub-samples), and each aliquot was assessed separately. Table 2.19 gives the results. Table 2.19 APC resistance split-samples data Compute the average CV over the 10 subjects as an overall measure of variability of the assay. In general, a CV of 10% is considered excellent, 10% and 20% is considered good, 20% and 30% is considered fair, and 30% is considered poor. How would you characterize the reliability of the APC assay based on these criteria?A study was conducted to demonstrate that soy beans inoculated with nitrogen-fixing bacteria yield more and grow adequately without expensive environmentally deleterious synthesized fertilizers. The trial was conducted under controlled conditions with uniform amounts of soil. The initial hypothesis was that inoculated plants would outperform their uninoculated counterparts. This assumption is based on the facts that plants need nitrogen to manufacture vital proteins and amino acids and that nitrogen-fixing bacteria would make more of this substance available to plants, increasing their size and yield. There were 8 inoculated plants (I) and 8 uninoculated plants (U). The plant yield as measured by pod weight for each plant is given in Table 2.20. Table 2.20 Pod weight (g) from inoculated (I) and uninoculated (U) plants Compute appropriate descriptive statistics for I and U plants.A study was conducted to demonstrate that soy beans inoculated with nitrogen-fixing bacteria yield more and grow adequately without expensive environmentally deleterious synthesized fertilizers. The trial was conducted under controlled conditions with uniform amounts of soil. The initial hypothesis was that inoculated plants would outperform their uninoculated counterparts. This assumption is based on the facts that plants need nitrogen to manufacture vital proteins and amino acids and that nitrogen-fixing bacteria would make more of this substance available to plants, increasing their size and yield. There were 8 inoculated plants (I) and 8 uninoculated plants (U). The plant yield as measured by pod weight for each plant is given in Table 2.20. Table 2.20 Pod weight (g) from inoculated (I) and uninoculated (U) plants Use graphic methods to compare the two groups.A study was conducted to demonstrate that soy beans inoculated with nitrogen-fixing bacteria yield more and grow adequately without expensive environmentally deleterious synthesized fertilizers. The trial was conducted under controlled conditions with uniform amounts of soil. The initial hypothesis was that inoculated plants would outperform their uninoculated counterparts. This assumption is based on the facts that plants need nitrogen to manufacture vital proteins and amino acids and that nitrogen-fixing bacteria would make more of this substance available to plants, increasing their size and yield. There were 8 inoculated plants (I) and 8 uninoculated plants (U). The plant yield as measured by pod weight for each plant is given in Table 2.20. What is your overall impression concerning the pod weight in the two groups?38PIn Section 2.10, we described Data Set BONEDEN.DAT (at www.cengagebrain.com) concerning the effect of tobacco use on BMD. Suppose we group the twin pairs according to the difference in tobacco use expressed in 10 pack-year groups (09.9 pack-years/1019.9 pack-years/2029.9 pack-years/3039.9 pack-years/40+ pack-years). Compute appropriate descriptive statistics, and provide a scatter plot for C grouped by the difference in tobacco use in pack-years.What impression do you have of the relationship between BMD and tobacco use based on Problem 2.39? 2.39 Suppose we group the twin pairs according to the difference in tobacco use expressed in 10 pack-year groups (09.9 pack-years/1019.9 pack-years/2029.9 pack-years/3039.9 pack-years/40+ pack-years). Compute appropriate descriptive statistics, and provide a scatter plot for C grouped by the difference in tobacco use in pack-years.In Section 2.10, we described Data Set BONEDEN.DAT (at www.cengagebrain.com) concerning the effect of tobacco use on BMD. Answer Problems 2.382.40 for BMD for the femoral neck. 2.38 For each pair of twins, compute the following for the lumbar spine: A = BMD for the heavier-smoking twin BMD for the lighter-smoking twin = X1 X2 B = mean BMD for the twinship = (X1 + X2)/2 C = 100% (A/B) Derive appropriate descriptive statistics for C over the entire study population.In Section 2.10, we described Data Set BONEDEN.DAT (at www.cengagebrain.com) concerning the effect of tobacco use on BMD. Answer Problems 2.382.40 for BMD for the femoral shaft. 2.39 Suppose we group the twin pairs according to the difference in tobacco use expressed in 10 pack-year groups (09.9 pack-years/1019.9 pack-years/2029.9 pack-years/3039.9 pack-years/40+ pack-years). Compute appropriate descriptive statistics, and provide a scatter plot for C grouped by the difference in tobacco use in pack-years.43P44P45PAnswer Problems 2.382.40 for BMD for the femoral shaft. 2.40 What impression do you have of the relationship between BMD and tobacco use based on Problem 2.39?The Left Ventricular Mass lndex (LVMI) is a measure of the enlargement of the left side of the heart and is expressed in the units (gm/ht(m)2.7). High values may predict subsequent cardiovascular disease in children as they get older (Urbina et al., [11]). A study is performed to relate the level of LVMI to blood pressure category in children and adolescents age 1018. The bp level of children was categorized as either Normal (bpcat = 1 or bp percentile 80% for a given age, gender, and height), Pre-Hypertensive (bpcat = 2 or bp percentile 80% and bp percentile 90%), or Hypertensive (bpcat = 3 or bp percentile 90%). The data are available in the data set LVM.XLS at www.cengagebrain.com What is the arithmetic mean of LVMI by blood pressure group?48P49PThe Left Ventricular Mass lndex (LVMI) is a measure of the enlargement of the left side of the heart and is expressed in the units (gm/ht(m)2.7). High values may predict subsequent cardiovascular disease in children as they get older (Urbina et al., [11]). A study is performed to relate the level of LVMI to blood pressure category in children and adolescents age 1018. The bp level of children was categorized as either Normal (bpcat = 1 or bp percentile 80% for a given age, gender, and height), Pre-Hypertensive (bpcat = 2 or bp percentile 80% and bp percentile 90%), or Hypertensive (bpcat = 3 or bp percentile 90%). The data are available in the data set LVM.XLS at www.cengagebrain.com Based on the box plot, does the arithmetic mean or the geometric mean provide a more appropriate measure of location for this type of data?When is it appropriate to use the arithmetic mean as opposed to the median?How does the geometric mean differ from the arithmetic mean? For what type of data is the geometric mean used?What is the difference between the standard deviation and the CV? When is it appropriate to use each measure?What is a stem-and-leaf plot? How does it differ from a bar graph?B.2REConsider the stem-and-leaf plot in Figure 2.6. Is it possible to construct a bar graph from the data presented? If so, construct the plot.What is a box plot? What additional information does this type of display give that is not available from either a bar graph or stem-and-leaf plot?Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}. 3.1 What does A1 A2 mean?Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}. What does A1 A2 mean?Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}. 3.3 Are A3 and A4 mutually exclusive?Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}. 3.4 What does A3 B mean?Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}. 3.5 What does A3 B mean?Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}. 3.6 Express C in terms of A1, A2, A3, and A4.Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}. 3.7 Express D in terms of B and C.Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}. 3.8 What does A1 mean?Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}. 3.9 What does A2 mean?Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}. 3.10 Represent C in terms of A1, A2, A3, and A4.11PSuppose an influenza epidemic strikes a city. In 10% of families the mother has influenza; in 10% of families the father has influenza; and in 2% of families both the mother and father have influenza. Are the events A1 = {mother has influenza} and A2 = {father has influenza} independent? Suppose there is a 20% chance each child will get influenza, whereas in 10% of two-child families both children get the disease.Suppose there is a 20% chance each child will get influenza, whereas in 10% of two-child families both children get the disease. What is the probability that at least one child will get influenza?Suppose there is a 20% chance each child will get influenza, whereas in 10% of two-child families both children get the disease. Based on Problem 3.12, what is the conditional probability that the father has influenza given that the mother has influenza?Suppose there is a 20% chance each child will get influenza, whereas in 10% of two-child families both children get the disease. Based on Problem 3.12, what is the conditional probability that the father has influenza given that the mother does not have influenza?Mental Health Estimates of the prevalence of Alzheimers disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table 3.5. Suppose an unrelated 77-year-old man, 76-year-old woman, and 82-year-old woman are selected from a community. What is the probability that all three of these individuals have Alzheimers disease? Table 3.5 Prevalence of Alzheimers disease (cases per 100 population)Mental Health Estimates of the prevalence of Alzheimers disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table 3.5. Suppose an unrelated 77-year-old man, 76-year-old woman, and 82-year-old woman are selected from a community. Table 3.5 Prevalence of Alzheimers disease (cases per 100 population) What is the probability that at least one of the women has Alzheimers disease?Mental Health Estimates of the prevalence of Alzheimers disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table 3.5. Suppose an unrelated 77-year-old man, 76-year-old woman, and 82-year-old woman are selected from a community. Table 3.5 Prevalence of Alzheimers disease (cases per 100 population) What is the probability that at least one of the three people has Alzheimers disease?Mental Health Estimates of the prevalence of Alzheimers disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table 3.5. Suppose an unrelated 77-year-old man, 76-year-old woman, and 82-year-old woman are selected from a community. Table 3.5 Prevalence of Alzheimers disease (cases per 100 population) What is the probability that exactly one of the three people has Alzheimers disease?Estimates of the prevalence of Alzheimers disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table 3.5. Suppose an unrelated 77-year-old man, 76-year-old woman, and 82-year-old woman are selected from a community. Suppose we know one of the three people has Alzheimers disease, but we dont know which one. What is the conditional probability that the affected person is a woman? Table 3.5 Prevalence of Alzheimers disease (cases per 100 population)Estimates of the prevalence of Alzheimers disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table 3.5. Suppose an unrelated 77-year-old man, 76-year-old woman, and 82-year-old woman are selected from a community. Suppose we know two of the three people have Alzheimers disease. What is the conditional probability that they are both women? Table 3.5 Prevalence of Alzheimers disease (cases per 100 population)Estimates of the prevalence of Alzheimers disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table 3.5. Suppose an unrelated 77-year-old man, 76-year-old woman, and 82-year-old woman are selected from a community. Suppose we know two of the three people have Alzheimers disease. What is the conditional probability that they are both younger than 80 years of age? Table 3.5 Prevalence of Alzheimers disease (cases per 100 population)Estimates of the prevalence of Alzheimers disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table 3.5. Suppose the probability that both members of a married couple, each of whom is 7579 years of age, will have Alzheimers disease is .0015. What is the conditional probability that the man will be affected given that the woman is affected? How does this value compare with the prevalence in Table 3.5? Why should it be the same (or different)? Table 3.5 Prevalence of Alzheimers disease (cases per 100 population)Estimates of the prevalence of Alzheimers disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table 3.5. Suppose the probability that both members of a married couple, each of whom is 7579 years of age, will have Alzheimers disease is .0015. What is the conditional probability that the woman will be affected given that the man is affected? How does this value compare with the prevalence in Table 3.5? Why should it be the same (or different)? Table 3.5 Prevalence of Alzheimers disease (cases per 100 population)Estimates of the prevalence of Alzheimers disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table 3.5. Suppose the probability that both members of a married couple, each of whom is 7579 years of age, will have Alzheimers disease is .0015. What is the probability that at least one member of the couple is affected? Suppose a study of Alzheimers disease is proposed in a retirement community with people 65+ years of age, where the agegender distribution is as shown in Table 3.6. Table 3.5 Prevalence of Alzheimers disease (cases per 100 population)Suppose a study of Alzheimers disease is proposed in a retirement community with people 65+ years of age, where the agegender distribution is as shown in Table 3.6. Table 3.6 Agegender distribution of retirement community aPercentage of total population. What is the expected overall prevalence of Alzheimers disease in the community if the prevalence estimates in Table 3.5 for specific agegender groups hold? Table 3.5 Prevalence of Alzheimers disease (cases per 100 population)Suppose a study of Alzheimers disease is proposed in a retirement community with people 65+ years of age, where the agegender distribution is as shown in Table 3.6. Table 3.6 Agegender distribution of retirement community aPercentage of total population. Assuming there are 1000 people 65+ years of age in the community, what is the expected number of cases of Alzheimers disease in the community?Commonly used vaccines for influenza are trivalent and contain only one type of influenza B virus. They may be ineffective against other types of influenza B virus. A randomized clinical trial was performed among children 3 to 8 years of age in 8 countries. Children received either a quadrivalent vaccine (QIV) that had more than one influenza B virus or a trivalent Hepatitis A vaccine (control) (Jain, et al., [9]. New England Journal of Medicine 2013: 369(26): 24812491). An attack rate (i.e.,% of children who developed influenza) starting 14 days after vaccination until the end of the study was computed for each vaccine group, stratified by age. The following data were reported: Table 3.7 Attack rate for influenza by age and treatment group Suppose 3 children in a village ages 3, 5, and 7 are vaccinated with the QIV vaccine. What is the probability that at least one child among the 3 will get influenza?29P30P31PGenetics Suppose that a disease is inherited via a dominant mode of inheritance and that only one of the two parents is affected with the disease. The implications of this mode of inheritance are that the probability is 1 in 2 that any particular offspring will get the disease. What is the probability that in a family with two children, both siblings are affected?Genetics Suppose that a disease is inherited via a dominant mode of inheritance and that only one of the two parents is affected with the disease. The implications of this mode of inheritance are that the probability is 1 in 2 that any particular offspring will get the disease. What is the probability that exactly one sibling is affected?Genetics Suppose that a disease is inherited via a dominant mode of inheritance and that only one of the two parents is affected with the disease. The implications of this mode of inheritance are that the probability is 1 in 2 that any particular offspring will get the disease. What is the probability that neither sibling is affected?35P36PSuppose that a disease is inherited via an autosomal recessive mode of inheritance. The implications of this mode of inheritance are that the children in a family each have a probability of 1 in 4 of inheriting the disease. What is the probability that in a family with two children, both siblings are affected?Suppose that a disease is inherited via an autosomal recessive mode of inheritance. The implications of this mode of inheritance are that the children in a family each have a probability of 1 in 4 of inheriting the disease. What is the probability that exactly one sibling is affected?39PSuppose that a disease is inherited via a sex-linked mode of inheritance. The implications of this mode of inheritance are that each male offspring has a 50% chance of inheriting the disease, whereas the female offspring have no chance of getting the disease. In a family with one male and one female sibling, what is the probability that both siblings are affected?Suppose that a disease is inherited via a sex-linked mode of inheritance. The implications of this mode of inheritance are that each male offspring has a 50% chance of inheriting the disease, whereas the female offspring have no chance of getting the disease. What is the probability that exactly one sibling is affected?Suppose that a disease is inherited via a sex-linked mode of inheritance. The implications of this mode of inheritance are that each male offspring has a 50% chance of inheriting the disease, whereas the female offspring have no chance of getting the disease. What is the probability that neither sibling is affected?Suppose that a disease is inherited via a sex-linked mode of inheritance. The implications of this mode of inheritance are that each male offspring has a 50% chance of inheriting the disease, whereas the female offspring have no chance of getting the disease. Answer Problem 3.40 for families with two male siblings. 3.40 In a family with one male and one female sibling, what is the probability that both siblings are affected?44P45P46P47P48P49PObstetrics The following data are derived from the Monthly Vital Statistics Report (October 1999) issued by the National Center for Health Statistics [10]. These data are pertinent to livebirths only. Suppose that infants are classified as low birthweight if they have a birthweight 2500 g and as normal birthweight if they have a birthweight 2500 g. Suppose that infants are also classified by length of gestation in the following five categories: 28 weeks, 2831 weeks, 3235 weeks, 36 weeks, and 37 weeks. Assume the probabilities of the different periods of gestation are as given in Table 3.8. Also assume that the probability of low birthweight is .949 given a gestation of 28 weeks, .702 given a gestation of 2831 weeks, .434 given a gestation of 3235 weeks, .201 given a gestation of 36 weeks, and .029 given a gestation of 37 weeks. What is the probability of having a low birthweight infant? Table 3.8 Distribution of length of gestationObstetrics The following data are derived from the Monthly Vital Statistics Report (October 1999) issued by the National Center for Health Statistics [10]. These data are pertinent to livebirths only. Suppose that infants are classified as low birthweight if they have a birthweight 2500 g and as normal birthweight if they have a birthweight 2500 g. Suppose that infants are also classified by length of gestation in the following five categories: 28 weeks, 2831 weeks, 3235 weeks, 36 weeks, and 37 weeks. Assume the probabilities of the different periods of gestation are as given in Table 3.8. Also assume that the probability of low birthweight is .949 given a gestation of 28 weeks, .702 given a gestation of 2831 weeks, .434 given a gestation of 3235 weeks, .201 given a gestation of 36 weeks, and .029 given a gestation of 37 weeks. Table 3.8 Distribution of length of gestation Show that the events {length of gestation 31 weeks} and {low birthweight} are not independent.Obstetrics The following data are derived from the Monthly Vital Statistics Report (October 1999) issued by the National Center for Health Statistics [10]. These data are pertinent to livebirths only. Suppose that infants are classified as low birthweight if they have a birthweight 2500 g and as normal birthweight if they have a birthweight 2500 g. Suppose that infants are also classified by length of gestation in the following five categories: 28 weeks, 2831 weeks, 3235 weeks, 36 weeks, and 37 weeks. Assume the probabilities of the different periods of gestation are as given in Table 3.8. Also assume that the probability of low birthweight is .949 given a gestation of 28 weeks, .702 given a gestation of 2831 weeks, .434 given a gestation of 3235 weeks, .201 given a gestation of 36 weeks, and .029 given a gestation of 37 weeks. Table 3.8 Distribution of length of gestation What is the probability of having a length of gestation 36 weeks given that an infant is low birthweight?Pulmonary Disease The familial aggregation of respiratory disease is a well-established clinical phenomenon. However, whether this aggregation is due to genetic or environmental factors or both is somewhat controversial. An investigator wishes to study a particular environmental factor, namely the relationship of cigarette-smoking habits in the parents to the presence or absence of asthma in their oldest child age 5 to 9 years living in the household (referred to below as their offspring). Suppose the investigator finds that (1) if both the mother and father are current smokers, then the probability of their offspring having asthma is .15; (2) if the mother is a current smoker and the father is not, then the probability of their offspring having asthma is .13; (3) if the father is a current smoker and the mother is not, then the probability of their offspring having asthma is .05; and (4) if neither parent is a current smoker, then the probability of their offspring having asthma is .04. Suppose the smoking habits of the parents are independent and the probability that the mother is a current smoker is .4, whereas the probability that the father is a current smoker is .5. What is the probability that both the father and mother are current smokers?Pulmonary Disease The familial aggregation of respiratory disease is a well-established clinical phenomenon. However, whether this aggregation is due to genetic or environmental factors or both is somewhat controversial. An investigator wishes to study a particular environmental factor, namely the relationship of cigarette-smoking habits in the parents to the presence or absence of asthma in their oldest child age 5 to 9 years living in the household (referred to below as their offspring). Suppose the investigator finds that (1) if both the mother and father are current smokers, then the probability of their offspring having asthma is .15; (2) if the mother is a current smoker and the father is not, then the probability of their offspring having asthma is .13; (3) if the father is a current smoker and the mother is not, then the probability of their offspring having asthma is .05; and (4) if neither parent is a current smoker, then the probability of their offspring having asthma is .04. Consider the subgroup of families in which the mother is not a current smoker. What is the probability that the father is a current smoker among such families? How does this probability differ from that calculated in Problem 3.53? 3.53 Suppose the smoking habits of the parents are independent and the probability that the mother is a current smoker is .4, whereas the probability that the father is a current smoker is .5. What is the probability that both the father and mother are current smokers?Suppose, alternatively, that if the father is a current smoker, then the probability that the mother is a current smoker is .6; whereas if the father is not a current smoker, then the probability that the mother is a current smoker is .2. Also assume that statements 1, 2, 3, and 4 above hold. If the probability that the father is a current smoker is .5, what is the probability that the father is a current smoker and that the mother is not a current smoker?Suppose, alternatively, that if the father is a current smoker, then the probability that the mother is a current smoker is .6; whereas if the father is not a current smoker, then the probability that the mother is a current smoker is .2. Also assume that statements 1, 2, 3, and 4 above hold. Are the current smoking habits of the father and the mother independent? Why or why not?57PThe familial aggregation of respiratory disease is a well-established clinical phenomenon. However, whether this aggregation is due to genetic or environmental factors or both is somewhat controversial. An investigator wishes to study a particular environmental factor, namely the relationship of cigarette-smoking habits in the parents to the presence or absence of asthma in their oldest child age 5 to 9 years living in the household (referred to below as their offspring). Suppose the investigator finds that (1) if both the mother and father are current smokers, then the probability of their offspring having asthma is .15; (2) if the mother is a current smoker and the father is not, then the probability of their offspring having asthma is .13; (3) if the father is a current smoker and the mother is not, then the probability of their offspring having asthma is .05; and (4) if neither parent is a current smoker, then the probability of their offspring having asthma is .04. Suppose, alternatively, that if the father is a current smoker, then the probability that the mother is a current smoker is .6; whereas if the father is not a current smoker, then the probability that the mother is a current smoker is .2. Also assume that statements 1, 2, 3, and 4 above hold. Suppose a child has asthma. What is the posterior probability that the father is a current smoker?The familial aggregation of respiratory disease is a well-established clinical phenomenon. However, whether this aggregation is due to genetic or environmental factors or both is somewhat controversial. An investigator wishes to study a particular environmental factor, namely the relationship of cigarette-smoking habits in the parents to the presence or absence of asthma in their oldest child age 5 to 9 years living in the household (referred to below as their offspring). Suppose the investigator finds that (1) if both the mother and father are current smokers, then the probability of their offspring having asthma is .15; (2) if the mother is a current smoker and the father is not, then the probability of their offspring having asthma is .13; (3) if the father is a current smoker and the mother is not, then the probability of their offspring having asthma is .05; and (4) if neither parent is a current smoker, then the probability of their offspring having asthma is .04. Suppose, alternatively, that if the father is a current smoker, then the probability that the mother is a current smoker is .6; whereas if the father is not a current smoker, then the probability that the mother is a current smoker is .2. Also assume that statements 1, 2, 3, and 4 above hold. What is the posterior probability that the mother is a current smoker if the child has asthma?The familial aggregation of respiratory disease is a well-established clinical phenomenon. However, whether this aggregation is due to genetic or environmental factors or both is somewhat controversial. An investigator wishes to study a particular environmental factor, namely the relationship of cigarette-smoking habits in the parents to the presence or absence of asthma in their oldest child age 5 to 9 years living in the household (referred to below as their offspring). Suppose the investigator finds that (1) if both the mother and father are current smokers, then the probability of their offspring having asthma is .15; (2) if the mother is a current smoker and the father is not, then the probability of their offspring having asthma is .13; (3) if the father is a current smoker and the mother is not, then the probability of their offspring having asthma is .05; and (4) if neither parent is a current smoker, then the probability of their offspring having asthma is .04. Suppose, alternatively, that if the father is a current smoker, then the probability that the mother is a current smoker is .6; whereas if the father is not a current smoker, then the probability that the mother is a current smoker is .2. Also assume that statements 1, 2, 3, and 4 above hold. Answer Problem 3.58 if the child does not have asthma. 3.58 Suppose a child has asthma. What is the posterior probability that the father is a current smoker?The familial aggregation of respiratory disease is a well-established clinical phenomenon. However, whether this aggregation is due to genetic or environmental factors or both is somewhat controversial. An investigator wishes to study a particular environmental factor, namely the relationship of cigarette-smoking habits in the parents to the presence or absence of asthma in their oldest child age 5 to 9 years living in the household (referred to below as their offspring). Suppose the investigator finds that (1) if both the mother and father are current smokers, then the probability of their offspring having asthma is .15; (2) if the mother is a current smoker and the father is not, then the probability of their offspring having asthma is .13; (3) if the father is a current smoker and the mother is not, then the probability of their offspring having asthma is .05; and (4) if neither parent is a current smoker, then the probability of their offspring having asthma is .04. Suppose, alternatively, that if the father is a current smoker, then the probability that the mother is a current smoker is .6; whereas if the father is not a current smoker, then the probability that the mother is a current smoker is .2. Also assume that statements 1, 2, 3, and 4 above hold. Answer Problem 3.59 if the child does not have asthma. 3.59 What is the posterior probability that the mother is a current smoker if the child has asthma?Pulmonary Disease The familial aggregation of respiratory disease is a well-established clinical phenomenon. However, whether this aggregation is due to genetic or environmental factors or both is somewhat controversial. An investigator wishes to study a particular environmental factor, namely the relationship of cigarette-smoking habits in the parents to the presence or absence of asthma in their oldest child age 5 to 9 years living in the household (referred to below as their offspring). Suppose the investigator finds that (1) if both the mother and father are current smokers, then the probability of their offspring having asthma is .15; (2) if the mother is a current smoker and the father is not, then the probability of their offspring having asthma is .13; (3) if the father is a current smoker and the mother is not, then the probability of their offspring having asthma is .05; and (4) if neither parent is a current smoker, then the probability of their offspring having asthma is .04. Suppose, alternatively, that if the father is a current smoker, then the probability that the mother is a current smoker is .6; whereas if the father is not a current smoker, then the probability that the mother is a current smoker is .2. Also assume that statements 1, 2, 3, and 4 above hold. Are the childs asthma status and the fathers smoking status independent? Why or why not?Pulmonary Disease The familial aggregation of respiratory disease is a well-established clinical phenomenon. However, whether this aggregation is due to genetic or environmental factors or both is somewhat controversial. An investigator wishes to study a particular environmental factor, namely the relationship of cigarette-smoking habits in the parents to the presence or absence of asthma in their oldest child age 5 to 9 years living in the household (referred to below as their offspring). Suppose the investigator finds that (1) if both the mother and father are current smokers, then the probability of their offspring having asthma is .15; (2) if the mother is a current smoker and the father is not, then the probability of their offspring having asthma is .13; (3) if the father is a current smoker and the mother is not, then the probability of their offspring having asthma is .05; and (4) if neither parent is a current smoker, then the probability of their offspring having asthma is .04. Suppose, alternatively, that if the father is a current smoker, then the probability that the mother is a current smoker is .6; whereas if the father is not a current smoker, then the probability that the mother is a current smoker is .2. Also assume that statements 1, 2, 3, and 4 above hold. Are the childs asthma status and the mothers smoking status independent? Why or why not?Genetics, Obstetrics Precise quantification of smoking during pregnancy is difficult in retrospective studies. Routinely collected blood specimens from newborns for screening purposes may provide a low-cost method to objectively measure maternal smoking close to the time of delivery. Serum cotinine is an important biomarker of recent smoking. A study was performed comparing cotinine levels in dried blood spots in newborns with those in umbilical cord blood (the gold standard) among 428 newborns in the California Genetic Screening Program (Yang et al. [11]). The lowest detection limit for dried blood spot cotinine was 3.1 ng/mL. The data in Table 3.9 were presented relating dried blood spot cotinine determinations to umbilical cord blood cotinine determinations. Suppose a cutoff of 5 ng/mL is proposed as a criterion for testing positive based on dried blood spot cotinine levels. What is the sensitivity using this cut-point? TABLE 3.9 Distribution of Cotinine Level in Dried Blood Spots from Newborns by Maternal Active Smoking Status close to the time of delivery among 428 babies delivered in California, 20012003Precise quantification of smoking during pregnancy is difficult in retrospective studies. Routinely collected blood specimens from newborns for screening purposes may provide a low-cost method to objectively measure maternal smoking close to the time of delivery. Serum cotinine is an important biomarker of recent smoking. A study was performed comparing cotinine levels in dried blood spots in newborns with those in umbilical cord blood (the gold standard) among 428 newborns in the California Genetic Screening Program (Yang et al. [11]). The lowest detection limit for dried blood spot cotinine was 3.1 ng/mL. The data in Table 3.9 were presented relating dried blood spot cotinine determinations to umbilical cord blood cotinine determinations. Suppose a cutoff of 5 ng/mL is proposed as a criterion for testing positive based on dried blood spot cotinine levels. TABLE 3.9 Distribution of Cotinine Level in Dried Blood Spots from Newborns by Maternal Active Smoking Status close to the time of delivery among 428 babies delivered in California, 20012003 What is the specificity using this cut-point?Precise quantification of smoking during pregnancy is difficult in retrospective studies. Routinely collected blood specimens from newborns for screening purposes may provide a low-cost method to objectively measure maternal smoking close to the time of delivery. Serum cotinine is an important biomarker of recent smoking. A study was performed comparing cotinine levels in dried blood spots in newborns with those in umbilical cord blood (the gold standard) among 428 newborns in the California Genetic Screening Program (Yang et al. [11]). The lowest detection limit for dried blood spot cotinine was 3.1 ng/mL. The data in Table 3.9 were presented relating dried blood spot cotinine determinations to umbilical cord blood cotinine determinations. Suppose a cutoff of 5 ng/mL is proposed as a criterion for testing positive based on dried blood spot cotinine levels. TABLE 3.9 Distribution of Cotinine Level in Dried Blood Spots from Newborns by Maternal Active Smoking Status close to the time of delivery among 428 babies delivered in California, 20012003 Maternal active smoking at the time of delivery was defined as cord blood levels of 10 ng/mL. Suppose it is estimated based on a large sample of births in California that 20% of mothers smoke at the time of delivery. Suppose the screening test for detecting whether a mother smokes at the time of pregnancy is based on a cutoff of 5 ng/mL using dried blood specimens from the newborn. What is the probability that a mother smokes at the time of delivery if the dried blood specimen is 5 ng/mL?What is another name for this quantity? Suppose it is estimated based on a large sample of births in California that 20% of mothers smoke at the time of delivery. Suppose the screening test for detecting whether a mother smokes at the time of pregnancy is based on a cutoff of 5 ng/mL using dried blood specimens from the newborn. 3.66 What is the probability that a mother smokes at the time of delivery if the dried blood specimen is 5 ng/mL?Research into cigarette-smoking habits, smoking prevention, and cessation programs necessitates accurate measurement of smoking behavior. However, decreasing social acceptability of smoking appears to cause significant underreporting. Chemical markers for cigarette use can provide objective indicators of smoking behavior. One widely used noninvasive marker is the level of saliva thiocyanate (SCN). In a Minneapolis school district, 1332 students in eighth grade (ages 1214) participated in a study [12] whereby they (1) Viewed a film illustrating how recent cigarette use could be readily detected from small samples of saliva (2) Provided a personal sample of SCN (3) Provided a self-report of the number of cigarettes smoked per week The results are given in Table 3.10. TABLE 3.10 Relationship between SCN levels and self-reported cigarettes smoked per week Source: Based on the American Journal of Public Health, 71(12), 1320, 1981. Suppose the self-reports are completely accurate and are representative of the number of eighth-grade students who smoke in the general community. We are considering using an SCN level 100 g/mL as a test criterion for identifying cigarette smokers. Regard a student as positive if he or she smokes one or more cigarettes per week. What is the sensitivity of the test for light-smoking students (students who smoke 14 cigarettes per week)?Research into cigarette-smoking habits, smoking prevention, and cessation programs necessitates accurate measurement of smoking behavior. However, decreasing social acceptability of smoking appears to cause significant underreporting. Chemical markers for cigarette use can provide objective indicators of smoking behavior. One widely used noninvasive marker is the level of saliva thiocyanate (SCN). In a Minneapolis school district, 1332 students in eighth grade (ages 1214) participated in a study [12] whereby they (1) Viewed a film illustrating how recent cigarette use could be readily detected from small samples of saliva (2) Provided a personal sample of SCN (3) Provided a self-report of the number of cigarettes smoked per week The results are given in Table 3.10. TABLE 3.10 Relationship between SCN levels and self-reported cigarettes smoked per week Source: Based on the American Journal of Public Health, 71(12), 1320, 1981. Suppose the self-reports are completely accurate and are representative of the number of eighth-grade students who smoke in the general community. We are considering using an SCN level 100 g/mL as a test criterion for identifying cigarette smokers. Regard a student as positive if he or she smokes one or more cigarettes per week. What is the sensitivity of the test for moderate-smoking students (students who smoke 1544 cigarettes per week)?Research into cigarette-smoking habits, smoking prevention, and cessation programs necessitates accurate measurement of smoking behavior. However, decreasing social acceptability of smoking appears to cause significant underreporting. Chemical markers for cigarette use can provide objective indicators of smoking behavior. One widely used noninvasive marker is the level of saliva thiocyanate (SCN). In a Minneapolis school district, 1332 students in eighth grade (ages 1214) participated in a study [12] whereby they (1) Viewed a film illustrating how recent cigarette use could be readily detected from small samples of saliva (2) Provided a personal sample of SCN (3) Provided a self-report of the number of cigarettes smoked per week The results are given in Table 3.10. TABLE 3.10 Relationship between SCN levels and self-reported cigarettes smoked per week Source: Based on the American Journal of Public Health, 71(12), 1320, 1981. Suppose the self-reports are completely accurate and are representative of the number of eighth-grade students who smoke in the general community. We are considering using an SCN level 100 g/mL as a test criterion for identifying cigarette smokers. Regard a student as positive if he or she smokes one or more cigarettes per week. What is the sensitivity of the test for heavy-smoking students (students who smoke 45 cigarettes per week)?Research into cigarette-smoking habits, smoking prevention, and cessation programs necessitates accurate measurement of smoking behavior. However, decreasing social acceptability of smoking appears to cause significant underreporting. Chemical markers for cigarette use can provide objective indicators of smoking behavior. One widely used noninvasive marker is the level of saliva thiocyanate (SCN). In a Minneapolis school district, 1332 students in eighth grade (ages 1214) participated in a study [12] whereby they (1) Viewed a film illustrating how recent cigarette use could be readily detected from small samples of saliva (2) Provided a personal sample of SCN (3) Provided a self-report of the number of cigarettes smoked per week The results are given in Table 3.10. TABLE 3.10 Relationship between SCN levels and self-reported cigarettes smoked per week Source: Based on the American Journal of Public Health, 71(12), 1320, 1981. Suppose the self-reports are completely accurate and are representative of the number of eighth-grade students who smoke in the general community. We are considering using an SCN level 100 g/mL as a test criterion for identifying cigarette smokers. Regard a student as positive if he or she smokes one or more cigarettes per week. What is the specificity of the test?Research into cigarette-smoking habits, smoking prevention, and cessation programs necessitates accurate measurement of smoking behavior. However, decreasing social acceptability of smoking appears to cause significant underreporting. Chemical markers for cigarette use can provide objective indicators of smoking behavior. One widely used noninvasive marker is the level of saliva thiocyanate (SCN). In a Minneapolis school district, 1332 students in eighth grade (ages 1214) participated in a study [12] whereby they (1) Viewed a film illustrating how recent cigarette use could be readily detected from small samples of saliva (2) Provided a personal sample of SCN (3) Provided a self-report of the number of cigarettes smoked per week The results are given in Table 3.10. TABLE 3.10 Relationship between SCN levels and self-reported cigarettes smoked per week Source: Based on the American Journal of Public Health, 71(12), 1320, 1981. Suppose the self-reports are completely accurate and are representative of the number of eighth-grade students who smoke in the general community. We are considering using an SCN level 100 g/mL as a test criterion for identifying cigarette smokers. Regard a student as positive if he or she smokes one or more cigarettes per week. What is the PV+ of the test?Research into cigarette-smoking habits, smoking prevention, and cessation programs necessitates accurate measurement of smoking behavior. However, decreasing social acceptability of smoking appears to cause significant underreporting. Chemical markers for cigarette use can provide objective indicators of smoking behavior. One widely used noninvasive marker is the level of saliva thiocyanate (SCN). In a Minneapolis school district, 1332 students in eighth grade (ages 1214) participated in a study [12] whereby they (1) Viewed a film illustrating how recent cigarette use could be readily detected from small samples of saliva (2) Provided a personal sample of SCN (3) Provided a self-report of the number of cigarettes smoked per week The results are given in Table 3.10. TABLE 3.10 Relationship between SCN levels and self-reported cigarettes smoked per week Source: Based on the American Journal of Public Health, 71(12), 1320, 1981. Suppose the self-reports are completely accurate and are representative of the number of eighth-grade students who smoke in the general community. We are considering using an SCN level 100 g/mL as a test criterion for identifying cigarette smokers. Regard a student as positive if he or she smokes one or more cigarettes per week. What is the PV of the test?Pulmonary Disease Research into cigarette-smoking habits, smoking prevention, and cessation programs necessitates accurate measurement of smoking behavior. However, decreasing social acceptability of smoking appears to cause significant underreporting. Chemical markers for cigarette use can provide objective indicators of smoking behavior. One widely used noninvasive marker is the level of saliva thiocyanate (SCN). In a Minneapolis school district, 1332 students in eighth grade (ages 1214) participated in a study [12] whereby they (1) Viewed a film illustrating how recent cigarette use could be readily detected from small samples of saliva (2) Provided a personal sample of SCN (3) Provided a self-report of the number of cigarettes smoked per week The results are given in Table 3.10. Table 3.10 Relationship between SCN levels and self-reported cigarettes smoked per week Suppose the self-reports are completely accurate and are representative of the number of eighth-grade students who smoke in the general community. We are considering using an SCN level 100 g/mL as a test criterion for identifying cigarette smokers. Regard a student as positive if he or she smokes one or more cigarettes per week. Suppose we regard the self-reports of all students who report some cigarette consumption as valid but estimate that 20% of students who report no cigarette consumption actually smoke 14 cigarettes per week and an additional 10% smoke 514 cigarettes per week. Assuming the percentage of students with SCN 100 g/mL in these two subgroups is the same as in those who truly report 14 and 514 cigarettes per week, compute the specificity under these assumptions.Pulmonary Disease Research into cigarette-smoking habits, smoking prevention, and cessation programs necessitates accurate measurement of smoking behavior. However, decreasing social acceptability of smoking appears to cause significant underreporting. Chemical markers for cigarette use can provide objective indicators of smoking behavior. One widely used noninvasive marker is the level of saliva thiocyanate (SCN). In a Minneapolis school district, 1332 students in eighth grade (ages 1214) participated in a study [12] whereby they (1) Viewed a film illustrating how recent cigarette use could be readily detected from small samples of saliva (2) Provided a personal sample of SCN (3) Provided a self-report of the number of cigarettes smoked per week The results are given in Table 3.10. Table 3.10 Relationship between SCN levels and self-reported cigarettes smoked per week Suppose the self-reports are completely accurate and are representative of the number of eighth-grade students who smoke in the general community. We are considering using an SCN level 100 g/mL as a test criterion for identifying cigarette smokers. Regard a student as positive if he or she smokes one or more cigarettes per week. Suppose we regard the self-reports of all students who report some cigarette consumption as valid but estimate that 20% of students who report no cigarette consumption actually smoke 14 cigarettes per week and an additional 10% smoke 514 cigarettes per week. Compute the PV under these altered assumptions. How does the true PV using a screening criterion of SCN 100 g/mL for identifying smokers compare with the PV based on self-reports obtained in Problem 3.73? 3.73 What is the PV of the test?Laboratory measures of cardiovascular reactivity are receiving increasing attention. Much of the expanded interest is based on the belief that these measures, obtained under challenge from physical and psychological stressors, may yield a more biologically meaningful index of cardiovascular function than more traditional static measures. Typically, measurement of cardiovascular reactivity involves the use of an automated blood-pressure monitor to examine the changes in blood pressure before and after a stimulating experience (such as playing a video game). For this purpose, blood-pressure measurements were made with the Vita-Stat blood-pressure machine both before and after playing a video game. Similar measurements were obtained using manual methods for measuring blood pressure. A person was classified as a reactor if his or her DBP increased by 10 mm Hg or more after playing the game and as a nonreactor otherwise. The results are given in Table 3.11. TABLE 3.11 Classification of cardiovascular reactivity using an automated and a manual sphygmomanometer If the manual measurements are regarded as the true measure of reactivity, then what is the sensitivity of automated DBP measurements?Laboratory measures of cardiovascular reactivity are receiving increasing attention. Much of the expanded interest is based on the belief that these measures, obtained under challenge from physical and psychological stressors, may yield a more biologically meaningful index of cardiovascular function than more traditional static measures. Typically, measurement of cardiovascular reactivity involves the use of an automated blood-pressure monitor to examine the changes in blood pressure before and after a stimulating experience (such as playing a video game). For this purpose, blood-pressure measurements were made with the Vita-Stat blood-pressure machine both before and after playing a video game. Similar measurements were obtained using manual methods for measuring blood pressure. A person was classified as a reactor if his or her DBP increased by 10 mm Hg or more after playing the game and as a nonreactor otherwise. The results are given in Table 3.11. TABLE 3.11 Classification of cardiovascular reactivity using an automated and a manual sphygmomanometer What is the specificity of automated DBP measurements?Laboratory measures of cardiovascular reactivity are receiving increasing attention. Much of the expanded interest is based on the belief that these measures, obtained under challenge from physical and psychological stressors, may yield a more biologically meaningful index of cardiovascular function than more traditional static measures. Typically, measurement of cardiovascular reactivity involves the use of an automated blood-pressure monitor to examine the changes in blood pressure before and after a stimulating experience (such as playing a video game). For this purpose, blood-pressure measurements were made with the Vita-Stat blood-pressure machine both before and after playing a video game. Similar measurements were obtained using manual methods for measuring blood pressure. A person was classified as a reactor if his or her DBP increased by 10 mm Hg or more after playing the game and as a nonreactor otherwise. The results are given in Table 3.11. TABLE 3.11 Classification of cardiovascular reactivity using an automated and a manual sphygmomanometer If the population tested is representative of the general population, then what are the PV+ and PV using this test?The data set in Table 3.12 is based on 214 children with acute otitis media (otitis media with effusion, or OME) who participated in a randomized clinical trial [13], Each child had OME at the beginning of the study in either one (unilateral cases) or both (bilateral cases) ears and was randomly assigned to receive a 14-day course of one of two antibiotics, either cefaclor (CEF) or amoxicillin (AMO). The data here concern the 203 children whose middle-ear status was determined during a 14-day follow-up visit. The data in Table 3.12 are presented in data set EAR.DAT (at www .cengagebrain.com). TABLE 3.12 Format for EAR.DAT ColumnVariableFormat or code 1-3ID 5 Clearance by 14 days1 = yes/0 = no 7 Antibiotic1 = CEF/2 = AMO 9 Age1 = 2 yrs/2 = 2-5 yrs 3=6+ yrs 11 Ear1 = 1st ear/2 = 2nd ear Does there seem to be any difference in the effect of the antibiotics on clearance of otitis media? Express your results in terms of relative risk (RR). Consider separate analyses for unilateral and bilateral cases. Also consider an analysis combining the two types of cases.The data set in Table 3.12 is based on 214 children with acute otitis media (otitis media with effusion, or OME) who participated in a randomized clinical trial [13], Each child had OME at the beginning of the study in either one (unilateral cases) or both (bilateral cases) ears and was randomly assigned to receive a 14-day course of one of two antibiotics, either cefaclor (CEF) or amoxicillin (AMO). The data here concern the 203 children whose middle-ear status was determined during a 14-day follow-up visit. The data in Table 3.12 are presented in data set EAR.DAT (at www I .cengagebrain.com). The investigators recorded the ages of the children because they felt this might be an important factor in determining outcome. Were they right? Try to express your results in terms of RR.The data set in Table 3.12 is based on 214 children with acute otitis media (otitis media with effusion, or OME) who participated in a randomized clinical trial [13], Each child had OME at the beginning of the study in either one (unilateral cases) or both (bilateral cases) ears and was randomly assigned to receive a 14-day course of one of two antibiotics, either cefaclor (CEF) or amoxicillin (AMO). The data here concern the 203 children whose middle-ear status was determined during a 14-day follow-up visit. The data in Table 3.12 are presented in data set EAR.DAT (at www .cengagebrain.com). While controlling for age, propose an analysis com- paring the effectiveness of the two antibiotics. Express your results in terms of RR.A drug company is developing a new pregnancy-test kit for use on an outpatient basis. The company uses the pregnancy test on 100 women who are known to be pregnant, for whom 95 test results are positive. The company uses the pregnancy test on 100 other women who are known to not be pregnant, of whom 99 test negative. What is the sensitivity of the test?A drug company is developing a new pregnancy-test kit for use on an outpatient basis. The company uses the pregnancy test on 100 women who are known to be pregnant, for whom 95 test results are positive. The company uses the pregnancy test on 100 other women who are known to not be pregnant, of whom 99 test negative. What is the specificity of the test?A drug company is developing a new pregnancy-test kit for use on an outpatient basis. The company uses the pregnancy test on 100 women who are known to be pregnant, for whom 95 test results are positive. The company uses the pregnancy test on 100 other women who are known to not be pregnant, of whom 99 test negative. The company anticipates that of the women who will use the pregnancy-test kit, 10% will actually be pregnant. What is the PV+ of the test?Gynecology A drug company is developing a new pregnancy-test kit for use on an outpatient basis. The company uses the pregnancy test on 100 women who are known to be pregnant, for whom 95 test results are positive. The company uses the pregnancy test on 100 other women who are known to not be pregnant, of whom 99 test negative. The company anticipates that of the women who will use the pregnancy-test kit, 10% will actually be pregnant. Suppose the cost of a false negative (2c) is twice that of a false positive (c) (because for a false negative prenatal care would be delayed during the first trimester of pregnancy). If the standard home pregnancy-test kit (made by another drug company) has a sensitivity of .98 and a specificity of .98, then which test (the new or standard) will cost the least per woman using it in the general population and by how much?The Chinese Mini-Mental Status Test (CMMS) consists of 114 items intended to identify people with Alzheimers disease and senile dementia among people in China [14]. An extensive clinical evaluation of this instrument was performed, whereby participants were interviewed by psychiatrists and nurses and a definitive diagnosis of dementia was made. Table 3.13 shows the results obtained for the subgroup of people with at least some formal education. Suppose a cutoff value of 20 on the test is used to identify people with dementia. Table 3.13 Relationship of clinical dementia to outcome on the Chinese Mini-Mental Status Test What is the sensitivity of the test?The Chinese Mini-Mental Status Test (CMMS) consists of 114 items intended to identify people with Alzheimers disease and senile dementia among people in China [14]. An extensive clinical evaluation of this instrument was performed, whereby participants were interviewed by psychiatrists and nurses and a definitive diagnosis of dementia was made. Table 3.13 shows the results obtained for the subgroup of people with at least some formal education. Suppose a cutoff value of 20 on the test is used to identify people with dementia. Table 3.13 Relationship of clinical dementia to outcome on the Chinese Mini-Mental Status Test What is the specificity of the test?Mental Health The Chinese Mini-Mental Status Test (CMMS) consists of 114 items intended to identify people with Alzheimers disease and senile dementia among people in China [14]. An extensive clinical evaluation of this instrument was performed, whereby participants were interviewed by psychiatrists and nurses and a definitive diagnosis of dementia was made. Table 3.13 shows the results obtained for the subgroup of people with at least some formal education. TABLE 3.13 Relationship of clinical dementia to outcome on the Chinese Mini-Mental Status Test The cutoff value of 20 on the CMMS used to identify people with dementia is arbitrary. Suppose we consider changing the cutoff. What are the sensitivity and specificity if cutoffs of 5, 10, 15, 20, 25, or 30 are used? Make a table of your results.Construct an ROC curve based on the table constructed in Problem 3.89. Mental Health The Chinese Mini-Mental Status Test (CMMS) consists of 114 items intended to identify people with Alzheimers disease and senile dementia among people in China [14]. An extensive clinical evaluation of this instrument was performed, whereby participants were interviewed by psychiatrists and nurses and a definitive diagnosis of dementia was made. Table 3.13 shows the results obtained for the subgroup of people with at least some formal education. TABLE 3.13 Relationship of clinical dementia to outcome on the Chinese Mini-Mental Status Test The cutoff value of 20 on the CMMS used to identify people with dementia is arbitrary. Suppose we consider changing the cutoff. What are the sensitivity and specificity if cutoffs of 5, 10, 15, 20, 25, or 30 are used? Make a table of your results.Mental Health The Chinese Mini-Mental Status Test (CMMS) consists of 114 items intended to identify people with Alzheimers disease and senile dementia among people in China [14]. An extensive clinical evaluation of this instrument was performed, whereby participants were interviewed by psychiatrists and nurses and a definitive diagnosis of dementia was made. Table 3.13 shows the results obtained for the subgroup of people with at least some formal education. TABLE 3.13 Relationship of clinical dementia to outcome on the Chinese Mini-Mental Status Test Suppose we want both the sensitivity and specificity to be at least 70%. Use the ROC curve to identify the possible value(s) to use as the cutoff for identifying people with dementia, based on these criteria.Calculate the area under the ROC curve. Interpret what this area means in words in the context of this problem.Demography A study based on data collected from the Medical Birth Registry of Norway looked at fertility rates according to survival outcomes of previous births [15]. The data are presented in Table 3.14. Table 3.14 Relationship of fertility rates to survival outcome of previous births in Norway What is the probability of having a livebirth (L) at a second birth given that the outcome of the first pregnancy was a stillbirth (D), that is, death?94PDemography A study based on data collected from the Medical Birth Registry of Norway looked at fertility rates according to survival outcomes of previous births [15]. The data are presented in Table 3.14. Table 3.14 Relationship of fertility rates to survival outcome of previous births in Norway What is the probability of 0, 1, and 2+ additional pregnancies if the first birth was a stillbirth?Demography A study based on data collected from the Medical Birth Registry of Norway looked at fertility rates according to survival outcomes of previous births [15]. The data are presented in Table 3.14. Table 3.14 Relationship of fertility rates to survival outcome of previous births in Norway Answer Problem 3.95 if the first birth was a live birth. 3.95 What is the probability of 0, 1, and 2+ additional pregnancies if the first birth was a stillbirth?97PThe 4 allele of the gene encoding apolipoprotein E (APOE) is strongly associated with Alzheimers disease, but its value in making the diagnosis remains uncertain. A study was conducted among 2188 patients who were evaluated at autopsy for Alzheimers disease by previously established pathological criteria [16]. Patients were also evaluated clinically for the presence of Alzheimers disease. The data in Table 3.15 were presented. Suppose the pathological diagnosis is considered the gold standard for Alzheimers disease. Table 3.15 Relationship between clinical and pathological diagnoses of Alzheimers disease What is the specificity of the test?Mental Health The 4 allele of the gene encoding apolipoprotein E (APOE) is strongly associated with Alzheimers disease, but its value in making the diagnosis remains uncertain. A study was conducted among 2188 patients who were evaluated at autopsy for Alzheimers disease by previously established pathological criteria [16]. Patients were also evaluated clinically for the presence of Alzheimers disease. The data in Table 3.15 were presented. Suppose the pathological diagnosis is considered the gold standard for Alzheimers disease. To possibly improve on the diagnostic accuracy of the clinical diagnosis for Alzheimers disease, information on both the APOE genotype as well as the clinical diagnosis were considered. The data are presented in Table 3.16. Suppose we consider the combination of both a clinical diagnosis for Alzheimers disease and the presence of 1 4 allele as a screening test for Alzheimers disease. What is the sensitivity of this test? Table 3.15 Relationship between clinical and pathological diagnoses of Alzheimers disease Table 3.16 Influence of the APOE genotype in diagnosing Alzheimers disease (AD)Mental Health The 4 allele of the gene encoding apolipoprotein E (APOE) is strongly associated with Alzheimers disease, but its value in making the diagnosis remains uncertain. A study was conducted among 2188 patients who were evaluated at autopsy for Alzheimers disease by previously established pathological criteria [16]. Patients were also evaluated clinically for the presence of Alzheimers disease. The data in Table 3.15 were presented. Suppose the pathological diagnosis is considered the gold standard for Alzheimers disease. To possibly improve on the diagnostic accuracy of the clinical diagnosis for Alzheimers disease, information on both the APOE genotype as well as the clinical diagnosis were considered. The data are presented in Table 3.16. Suppose we consider the combination of both a clinical diagnosis for Alzheimers disease and the presence of 1 4 allele as a screening test for Alzheimers disease. Table 3.15 Relationship between clinical and pathological diagnoses of Alzheimers disease Table 3.16 Influence of the APOE genotype in diagnosing Alzheimers disease (AD) What is the specificity of this test?Cardiovascular Disease A fascinating subject of recent interest is the Hispanic paradox: Census data show that coronary heart disease (CHD) has a lower prevalence in Hispanic people than in non-Hispanic whites (NHW) based on health interviews of representative samples of people from different ethnic groups from the U.S. population, although the risk-factor profile of Hispanics is generally worse (more hypertension, diabetes, and obesity in this group than in NHW). To study this further, researchers looked at a group of 1000 Hispanic men ages 5064 from several counties in Texas who were free of CHD in 1990 and followed them for 5 years. They found that 100 of the men had developed CHD (either fatal cases or nonfatal cases in which the men survived a heart attack). Is the proportion 100 out of 1000 a prevalence rate, an incidence rate, or neither?Cardiovascular Disease A fascinating subject of recent interest is the Hispanic paradox: Census data show that coronary heart disease (CHD) has a lower prevalence in Hispanic people than in non-Hispanic whites (NHW) based on health interviews of representative samples of people from different ethnic groups from the U.S. population, although the risk-factor profile of Hispanics is generally worse (more hypertension, diabetes, and obesity in this group than in NHW). To study this further, researchers looked at a group of 1000 Hispanic men ages 5064 from several counties in Texas who were free of CHD in 1990 and followed them for 5 years. They found that 100 of the men had developed CHD (either fatal cases or nonfatal cases in which the men survived a heart attack). Given other surveys over the same time period among NHW in these counties, the researchers expected that the comparable rate of CHD for NHW would be 8%. Another important parameter in the epidemiology of CHD is the case-fatality rate (the proportion of people who die among those who have a heart attack). Among the 100 CHD cases ascertained among Hispanics, 50 were fatal. What is the expected proportion of Hispanic men who will be identified by health surveys as having a previous heart attack in the past 5 years (who are by definition survivors) if we assume that the proportion of men with more than one nonfatal heart attack is negligible? What is the comparable proportion for NHW men if the expected case-fatality rate is 20% among NHW men with CHD?Are these proportions prevalence rates, incidence rates, or neither? Do the results in this problem give insight into why the Hispanic paradox occurs (do Hispanic men truly have lower risk of CHD as government surveys would indicate)? Why or why not? Given other surveys over the same time period among NHW in these counties, the researchers expected that the comparable rate of CHD for NHW would be 8%. Another important parameter in the epidemiology of CHD is the case-fatality rate (the proportion of people who die among those who have a heart attack). Among the 100 CHD cases ascertained among Hispanics, 50 were fatal. 3.102 What is the expected proportion of Hispanic men who will be identified by health surveys as having a previous heart attack in the past 5 years (who are by definition survivors) if we assume that the proportion of men with more than one nonfatal heart attack is negligible? What is the comparable proportion for NHW men if the expected case-fatality rate is 20% among NHW men with CHD?104PA dominantly inherited genetic disease is identified over several generations of a large family. However, about half the families have dominant disease with complete penetrance, whereby if a parent is affected there is a 50% probability that any one offspring will be affected. Similarly, about half the families have dominant disease with reduced penetrance, whereby if a parent is affected there is a 25% probability that any one offspring will be affected. Suppose in a particular family one parent and two of the two offspring are affected. What is the probability that exactly two of the two offspring will be affected in a family with dominant disease with reduced penetrance?A dominantly inherited genetic disease is identified over several generations of a large family. However, about half the families have dominant disease with complete penetrance, whereby if a parent is affected there is a 50% probability that any one offspring will be affected. Similarly, about half the families have dominant disease with reduced penetrance, whereby if a parent is affected there is a 25% probability that any one offspring will be affected. Suppose in a particular family one parent and two of the two offspring are affected. What is the probability that the mode of transmission for this particular family is dominant with complete penetrance? Is this a prior probability or a posterior probability?107PInfectious Disease, Cardiovascular Disease A validation study is to be performed in a local hospital to check the accuracy of assessment of hospital-acquired infection (INF) following coronary bypass surgery (coronary-artery bypass graft, or CABG). In a given year the hospital performs 1100 CABG procedures. A Centers for Disease Control and Prevention (CDC) algorithm is currently used to categorize subjects as having INF. To validate this algorithm, all CDC+ subjects (N = 100) and a random sample of CDC subjects (N = 1000) will be ascertained by an infectious-disease (ID) fellow and a detailed investigation will be performed, including a chart review and documentation of antibiotic use. Assume the ID-fellows determination is correct. Suppose 100 CDC+ subjects are ascertained, of whom the ID fellow confirms 80. Because there are a large number of CDC subjects (1000), only a sample of 100 is studied, of whom the ID fellow confirms 90. What is the PV+ of the CDC algorithm?Infectious Disease, Cardiovascular Disease A validation study is to be performed in a local hospital to check the accuracy of assessment of hospital-acquired infection (INF) following coronary bypass surgery (coronary-artery bypass graft, or CABG). In a given year the hospital performs 1100 CABG procedures. A Centers for Disease Control and Prevention (CDC) algorithm is currently used to categorize subjects as having INF. To validate this algorithm, all CDC+ subjects (N = 100) and a random sample of CDC subjects (N = 1000) will be ascertained by an infectious-disease (ID) fellow and a detailed investigation will be performed, including a chart review and documentation of antibiotic use. Assume the ID-fellows determination is correct. Suppose 100 CDC+ subjects are ascertained, of whom the ID fellow confirms 80. Because there are a large number of CDC subjects (1000), only a sample of 100 is studied, of whom the ID fellow confirms 90. What is the PV of the CDC algorithm?Infectious Disease, Cardiovascular Disease A validation study is to be performed in a local hospital to check the accuracy of assessment of hospital-acquired infection (INF) following coronary bypass surgery (coronary-artery bypass graft, or CABG). In a given year the hospital performs 1100 CABG procedures. A Centers for Disease Control and Prevention (CDC) algorithm is currently used to categorize subjects as having INF. To validate this algorithm, all CDC+ subjects (N = 100) and a random sample of CDC subjects (N = 1000) will be ascertained by an infectious-disease (ID) fellow and a detailed investigation will be performed, including a chart review and documentation of antibiotic use. Assume the ID-fellows determination is correct. Suppose 100 CDC+ subjects are ascertained, of whom the ID fellow confirms 80. Because there are a large number of CDC subjects (1000), only a sample of 100 is studied, of whom the ID fellow confirms 90. What is the sensitivity of the CDC algorithm?114PGenetics Suppose a birth defect has a recessive form of inheritance. In a study population, the recessive gene (a) initially has a prevalence of 25%. A subject has the birth defect if both maternal and paternal genes are of type a. In the general population, what is the probability that an individual will have the birth defect, assuming that maternal and paternal genes are inherited independently?A further study finds that after 10 generations (200 years) a lot of inbreeding has taken place in the population. Two subpopulations (populations A and B), consisting of 30% and 70% of the general population, respectively, have formed. Within population A, prevalence of the recessive gene is 40%, whereas in population B it is 10%. Suppose that in 25% of marriages both people are from population A, in 65% both are from population B, and in 10% there is one partner from population A and one from population B. What is the probability of a birth defect in the next generation?A further study finds that after 10 generations (200 years) a lot of inbreeding has taken place in the population. Two subpopulations (populations A and B), consisting of 30% and 70% of the general population, respectively, have formed. Within population A, prevalence of the recessive gene is 40%, whereas in population B it is 10%. Suppose that a baby is born with a birth defect, but the babys ancestry is unknown. What is the posterior probability that the baby will have both parents from population A, both parents from population B, or mixed ancestry, respectively? (Hint: Use Bayes rule.)Orthopedics Piriformis syndrome is a pelvic condition that involves malfunction of the piriformis muscle (a deep buttock muscle), which often causes back and buttock pain with sciatica (pain radiating down the leg). An electrophysiologic test to detect piriformis syndrome involves measuring nerveconduction velocity (NCV) at two nerves in the leg (the tibial and peroneal nerves) with the leg flexed in a specific position. Increases in NCV in these nerves are often associated with piriformis syndrome. The resulting test, called the flexion abduction and internal rotation (FAIR) test, is positive if the average NCV in these nerves is delayed by 2+ seconds relative to normal. A small study compared the FAIR test results with patient self-reports of how they feel on a visual analog scale (VAS) of 010, with 0 indicating no pain and 10 very severe pain. The results were as shown in Table 3.17. Suppose physicians consider the FAIR test the gold standard, with a FAIR test result of 2 defined as a true positive and a FAIR test result of 2 defined as a true negative. Suppose a VAS of 4 is considered a good clinical response based on self-report (a test-negative) and a VAS of 5 is considered a bad clinical response (a test-positive). Table 3.17 FAIR test results on piriformis syndrome patients What is the sensitivity of the VAS?119POrthopedics Piriformis syndrome is a pelvic condition that involves malfunction of the piriformis muscle (a deep buttock muscle), which often causes back and buttock pain with sciatica (pain radiating down the leg). An electrophysiologic test to detect piriformis syndrome involves measuring nerveconduction velocity (NCV) at two nerves in the leg (the tibial and peroneal nerves) with the leg flexed in a specific position. Increases in NCV in these nerves are often associated with piriformis syndrome. The resulting test, called the flexion abduction and internal rotation (FAIR) test, is positive if the average NCV in these nerves is delayed by 2+ seconds relative to normal. Table 3.17 FAIR test results on piriformis syndrome patients A small study compared the FAIR test results with patient self-reports of how they feel on a visual analog scale (VAS) of 010, with 0 indicating no pain and 10 very severe pain. The results were as shown in Table 3.17. Suppose physicians consider the FAIR test the gold standard, with a FAIR test result of 2 defined as a true positive and a FAIR test result of 2 defined as a true negative. Suppose a VAS of 4 is considered a good clinical response based on self-report (a test-negative) and a VAS of 5 is considered a bad clinical response (a test-positive). The cutoff points of 5 for a VAS test-positive and 4 for a VAS test-negative are arbitrary. Compute and graph the ROC curve for the VAS test by varying the cutoff point for a test-positive. (Use the cutoff points VAS 0, VAS 3, VAS 5, VAS 7, and VAS 11 as possible criteria for test-positive.)121PCancer Breast cancer is considered largely a hormonal disease. An important hormone in breast-cancer research is estradiol. The data in Table 3.18 on serum estradiol levels were obtained from 213 breast-cancer cases and 432 agematched controls. All women were age 5059 years. Suppose a serum-estradiol level of 20+ pg/mL is proposed as a screening criterion for identifying breast-cancer cases. Table 3.18 Serum-estradiol data What is the sensitivity of this test?Cancer Breast cancer is considered largely a hormonal disease. An important hormone in breast-cancer research is estradiol. The data in Table 3.18 on serum estradiol levels were obtained from 213 breast-cancer cases and 432 agematched controls. All women were age 5059 years. Suppose a serum-estradiol level of 20+ pg/mL is proposed as a screening criterion for identifying breast-cancer cases. Table 3.18 Serum-estradiol data What is the specificity of this test?Cancer Breast cancer is considered largely a hormonal disease. An important hormone in breast-cancer research is estradiol. The data in Table 3.18 on serum estradiol levels were obtained from 213 breast-cancer cases and 432 agematched controls. All women were age 5059 years. Suppose a serum-estradiol level of 20+ pg/mL is proposed as a screening criterion for identifying breast-cancer cases. Table 3.18 Serum-estradiol data The preceding sample was selected to oversample cases. In the general population, the prevalence of breast cancer is about 2% among women 5059 years of age. What is the probability of breast cancer among 50- to 59-year-old women in the general population who have a serum-estradiol level of 20 pg/mL? What is another name for this quantity?125PCardiovascular Disease Mayo Clinic investigators have tracked coronary-heart-disease (CHD) mortality in Olmstead County, Minnesota, for the past 20 years[17]. Mayo Clinic physicians provided virtually all medical care to Olmstead County residents. Deaths from CHD were subdivided into those that occurred in hospital and those that occurred out of hospital. In-hospital death rates are thought to be influenced mainly by advances in medical care. Out-of-hospital death rates are thought to be influenced mainly by changes in risk-factor levels over time. For men, out-of-hospital CHD death rates were 280 cases per 100,000 men per year and in-hospital CHD death rates were 120 cases per 100,000 men per year in 1998. For women, out-of-hospital CHD death rates were 100 cases per 100,000 women per year; in-hospital CHD death rates were 40 cases per 100,000 women per year in 1998. The investigators reported that for both men and women, in hospital CHD death rates were declining at a rate of 5.3% per year, whereas out-of-hospital CHD death rates were declining by 1.8% per year. What is the expected overall CHD mortality rate in Olmstead County in 2015 if these trends continue?127P128P129P130P131P132PRadiology Mobile displays have the potential to increase the flexibility of consulting radiologists if they can be shown to be comparable to traditional display modalities. A study was performed comparing a mobile display iPad 2 with a larger liquid crystal display (LCD) for the diagnosis of tuberculosis (TB) on chest radiography (Abboud et al., [19]). De-identified images of 240 chest X-rays were transferred from a PACS workstation (LCD) to an iPad 2 tablet. The images were reviewed independently by 5 radiologists and were graded as positive or negative for TB on both the LCD and the iPad 2. The reviews occurred at different times to avoid recall bias. A database of 500 chest X-rays was created from TB screening films over a 4-month period. Of these, 200 cases originally interpreted as TB-negative and 40 cases originally interpreted as TB-positive were selected at random for study. The images were re-reviewed using both an LCD and an iPad 2 imaging display, albeit at different times. The results were as shown in Table 3.21. Table 3.21 Comparison of TB screening results using an LCD and iPad 2 display If we regard the LCD interpretation as the gold standard, then what is the specificity of the iPad 2 interpretation?Radiology Mobile displays have the potential to increase the flexibility of consulting radiologists if they can be shown to be comparable to traditional display modalities. A study was performed comparing a mobile display iPad 2 with a larger liquid crystal display (LCD) for the diagnosis of tuberculosis (TB) on chest radiography (Abboud et al., [19]). De-identified images of 240 chest X-rays were transferred from a PACS workstation (LCD) to an iPad 2 tablet. The images were reviewed independently by 5 radiologists and were graded as positive or negative for TB on both the LCD and the iPad 2. The reviews occurred at different times to avoid recall bias. A database of 500 chest X-rays was created from TB screening films over a 4-month period. Of these, 200 cases originally interpreted as TB-negative and 40 cases originally interpreted as TB-positive were selected at random for study. The images were re-reviewed using both an LCD and an iPad 2 imaging display, albeit at different times. The results were as shown in Table 3.21. Table 3.21 Comparison of TB screening results using an LCD and iPad 2 display The selection of images for this study was enriched to increase the number of images originally interpreted as positive. Suppose the underlying percentage of positive TB tests is 10% in a large sample of chest X-rays assessed by LCD. If a subject tests positive on an iPad 2 display, then what is the probability that he(she) will also test positive on the LCD?135P136P137PCardiovascular Disease The ankle-arm blood-pressure index (AAI) is defined as the ratio of ankle systolic blood pressure/arm systolic blood pressure and is used for the diagnosis of lower extremity arterial disease. A study was conducted to investigate whether the AAI can be used as a screening test for atherosclerotic diseases in general [20]. The subjects were 446 male workers in a copper smelter in Japan. Each subject had an AAI determination as well as an electrocardiogram (ECG). From the ECG, an S-T segment depression was defined as an S-T segment 0.1 mV below the baseline in at least 1 of 12 leads in a resting ECG. S-T segment depression is often used as one characterization of an abnormal ECG. The data in Table 3.22 were presented relating AAI to S-T segment depression. Table 3.22 Association between ankle-arm blood-pressure index (AAI) and S-T segment depression What is the PV+? (Hint: Assume that the subjects in this study are a random sample from the general population of Japan.)Cardiovascular Disease The ankle-arm blood-pressure index (AAI) is defined as the ratio of ankle systolic blood pressure/arm systolic blood pressure and is used for the diagnosis of lower extremity arterial disease. A study was conducted to investigate whether the AAI can be used as a screening test for atherosclerotic diseases in general [20]. The subjects were 446 male workers in a copper smelter in Japan. Each subject had an AAI determination as well as an electrocardiogram (ECG). From the ECG, an S-T segment depression was defined as an S-T segment 0.1 mV below the baseline in at least 1 of 12 leads in a resting ECG. S-T segment depression is often used as one characterization of an abnormal ECG. The data in Table 3.22 were presented relating AAI to S-T segment depression. Table 3.22 Association between ankle-arm blood-pressure index (AAI) and S-T segment depression What is the PV?Cardiovascular Disease The ankle-arm blood-pressure index (AAI) is defined as the ratio of ankle systolic blood pressure/arm systolic blood pressure and is used for the diagnosis of lower extremity arterial disease. A study was conducted to investigate whether the AAI can be used as a screening test for atherosclerotic diseases in general [20]. The subjects were 446 male workers in a copper smelter in Japan. Each subject had an AAI determination as well as an electrocardiogram (ECG). From the ECG, an S-T segment depression was defined as an S-T segment 0.1 mV below the baseline in at least 1 of 12 leads in a resting ECG. S-T segment depression is often used as one characterization of an abnormal ECG. The data in Table 3.22 were presented relating AAI to S-T segment depression. Table 3.22 Association between ankle-arm blood-pressure index (AAI) and S-T segment depression Suppose the reproducibility of the AAI test were improved using better technology. Would the sensitivity of the test increase, decrease, or remain the same? why?141P142P143PObstetrics, Health Promotion A study was performed to assess the accuracy of self-reported exposure to cigarette smoking in-utero. A comparison was made between daughters reports of smoking by their mothers during pregnancy with the mothers self-reports of their own smoking while pregnant with their daughters. The results were as shown in Table 3.23. Table 3.23 Relationship between mothers self-reports of smoking while pregnant and daughters reports of fetal smoke exposure which positive indicates smoking and negative indicates not smoking? Additional data on self-reported smoking indicate that the mother is not always completely accurate. Saliva cotinine is a biochemical marker that, if elevated, is a 100% accurate indication of recent smoking. Suppose if the mother states she is a nonsmoker during pregnancy that saliva cotinine is elevated 5% of the time, whereas if the mother states she is a smoker during pregnancy that saliva cotinine is elevated 97% of the time. Assume also that a daughter report adds no further information regarding the probability of an elevated cotinine level once the mothers self-report is known. What is the probability that the saliva cotinine level in the mother is not elevated during pregnancy if the daughter reports that the mother did not smoke in-utero?What is the frequency definition of probability?What is the difference between independent and dependent events?What are mutually exclusive events?What is the addition law of probability?What is conditional probability? How does it differ from unconditional probability?What is relative risk? How do you interpret it?B.1RESuppose the rate of type II diabetes mellitus (DM) in 40- to 59-year-olds is 7% among Caucasians, 10% among African Americans, 12% among Hispanics, and 5% among Asian Americans. Suppose the ethnic distribution in Houston, Texas, among 40- to 59-year-olds is 30% Caucasian, 25% African American, 40% Hispanic, and 5% Asian American. What is the overall probability of type II DM among 40- to 59-year-olds in Houston?What is the sensitivity and specificity of a screening test?What are the PV+ and PV of a screening test? How does PV differ from sensitivity and specificity?C.3REWhat is Bayes rule? How is it used?D.2RED.3RED.4RESuppose that of 25 students in a class, 5 are currently suffering from hay fever. Is the proportion 5 of 25 (20%) a measure of prevalence, incidence, or neither?Suppose 50 HIV-positive men are identified, 5 of whom develop AIDS over the next 2 years. Is the proportion 5 of 50 (10%) a measure of prevalence, incidence, or neither?Let X be the random variable representing the number of hypertensive adults in Example 3.12. Derive the probability-mass function for X.Let X be the random variable representing the number of hypertensive adults in Example 3.12. What is its expected value?Let X be the random variable representing the number of hypertensive adults in Example 3.12. What is its variance?Let X be the random variable representing the number of hypertensive adults in Example 3.12. What is its cumulative-distribution function?Suppose we want to check the accuracy of self-reported diagnoses of angina by getting further medical records on a subset of the cases. If we have 50 reported cases of angina and we want to select 5 for further review, then how many ways can we select these cases if order of selection matters?Suppose we want to check the accuracy of self-reported diagnoses of angina by getting further medical records on a subset of the cases. Answer Problem 4.5 assuming order of selection does not matter. 4.5 If we have 50 reported cases of angina and we want to select 5 for further review, then how many ways can we select these cases if order of selection matters?Evaluate (100),(101),,(1010).Evaluate 9!.Suppose 6 of 15 students in a grade-school class develop influenza, whereas 20% of grade-school students nationwide develop influenza. Is there evidence of an excessive number of cases in the class? That is, what is the probability of obtaining at least 6 cases in this class if the nationwide rate holds true?What is the expected number of students in the class who will develop influenza?What is the probability of obtaining exactly 6 events for a Poisson distribution with parameter = 4.0?What is the probability of obtaining at least 6 events for a Poisson distribution with parameter = 4.0?What is the expected value and variance for a Poisson distribution with parameter = 4.0?14PInfectious Disease Newborns were screened for human immunodeficiency virus (HIV) or acquired immunodeficiency syndrome (AIDS) in five Massachusetts hospitals. The data [8] are shown in Table 4.15. Table 4.15 Seroprevalence of HIV antibody in newborns blood samples, according to hospital category If 500 newborns are screened at the inner-city hospital, then what is the exact binomial probability of at least 5 HIV-positive test results?Infectious Disease Newborns were screened for human immunodeficiency virus (HIV) or acquired immunodeficiency syndrome (AIDS) in five Massachusetts hospitals. The data [8] are shown in Table 4.15. Table 4.15 Seroprevalence of HIV antibody in newborns blood samples, according to hospital category Answer Problems 4.14 and 4.15 using an approximation rather than an exact probability. 4.14 If 500 newborns are screened at the inner-city hospital, then what is the exact binomial probability of exactly 5 HIV-positive test results? 4.15 If 500 newborns are screened at the inner-city hospital, then what is the exact binomial probability of at least 5 HIV-positive test results?Infectious Disease Newborns were screened for human immunodeficiency virus (HIV) or acquired immunodeficiency syndrome (AIDS) in five Massachusetts hospitals. The data [8] are shown in Table 4.15. Table 4.15 Seroprevalence of HIV antibody in newborns blood samples, according to hospital category Answer Problem 4.14 for a mixed urban/suburban hospital (hospital C). 4.14 If 500 newborns are screened at the inner-city hospital, then what is the exact binomial probability of exactly 5 HIV-positive test results?Infectious Disease Newborns were screened for human immunodeficiency virus (HIV) or acquired immunodeficiency syndrome (AIDS) in five Massachusetts hospitals. The data [8] are shown in Table 4.15. Table 4.15 Seroprevalence of HIV antibody in newborns blood samples, according to hospital category Answer Problem 4.15 for a mixed urban/suburban hospital (hospital C). 4.15 If 500 newborns are screened at the inner-city hospital, then what is the exact binomial probability of at least 5 HIV-positive test results?19PInfectious Disease Newborns were screened for human immunodeficiency virus (HIV) or acquired immunodeficiency syndrome (AIDS) in five Massachusetts hospitals. The data [8] are shown in Table 4.15. Table 4.15 Seroprevalence of HIV antibody in newborns blood samples, according to hospital category Answer Problem 4.14 for a mixed suburban/rural hospital (hospital E). 4.14 If 500 newborns are screened at the inner-city hospital, then what is the exact binomial probability of exactly 5 HIV-positive test results?Infectious Disease Newborns were screened for human immunodeficiency virus (HIV) or acquired immunodeficiency syndrome (AIDS) in five Massachusetts hospitals. The data [8] are shown in Table 4.15. Table 4.15 Seroprevalence of HIV antibody in newborns blood samples, according to hospital category Answer Problem 4.15 for a mixed suburban/rural hospital (hospital E). 4.15 If 500 newborns are screened at the inner-city hospital, then what is the exact binomial probability of at least 5 HIV-positive test results?Infectious Disease Newborns were screened for human immunodeficiency virus (HIV) or acquired immunodeficiency syndrome (AIDS) in five Massachusetts hospitals. The data [8] are shown in Table 4.15. Table 4.15 Seroprevalence of HIV antibody in newborns blood samples, according to hospital category Answer Problem 4.16 for a mixed suburban/rural hospital (hospital E). 4.14 If 500 newborns are screened at the inner-city hospital, then what is the exact binomial probability of exactly 5 HIV-positive test results? 4.15 If 500 newborns are screened at the inner-city hospital, then what is the exact binomial probability of at least 5 HIV-positive test results? 4.16 Answer Problems 4.14 and 4.15 using an approximation rather than an exact probability. 4.20 Answer Problem 4.14 for a mixed suburban/rural hospital (hospital E). 4.21 Answer Problem 4.15 for a mixed suburban/rural hospital (hospital E).Suppose 10 gonorrhea cases are reported over a 3-month period among 10,000 people living in an urban county. The statewide incidence of gonorrhea is 50 per 100,000 over a 3-month period. Is the number of gonorrhea cases in this county unusual for this time period?Assume the number of episodes per year of otitis media, a common disease of the middle ear in early childhood, follows a Poisson distribution with parameter = 1.6 episodes per year. Find the probability of getting 3 or more episodes of otitis media in the first 2 years of life.Assume the number of episodes per year of otitis media, a common disease of the middle ear in early childhood, follows a Poisson distribution with parameter = 1.6 episodes per year. Find the probability of not getting any episodes of otitis media in the first year of life.An interesting question in pediatrics is whether the tendency for children to have many episodes of otitis media is inherited in a family. What is the probability that 2 siblings will both have 3 or more episodes of otitis media in the first 2 years of life?What is the probability that exactly 1 sibling will have 3 or more episodes in the first 2 years of life?An interesting question in pediatrics is whether the tendency for children to have many episodes of otitis media is inherited in a family. What is the probability that neither sibling will have 3 or more episodes in the first 2 years of life?What is the expected number of siblings in a 2-sibling family who will have 3 or more episodes in the first 2 years of life?A national study found that treating people appropriately for high blood pressure reduced their overall mortality by 20%. Treating people adequately for hypertension has been difficult because it is estimated that 50% of hypertensives do not know they have high blood pressure, 50% of those who do know are inadequately treated by their physicians, and 50% who are appropriately treated fail to follow this treatment by taking the right number of pills. What is the probability that among 10 true hypertensives at least 50% are being treated appropriately and are complying with this treatment?A national study found that treating people appropriately for high blood pressure reduced their overall mortality by 20%. Treating people adequately for hypertension has been difficult because it is estimated that 50% of hypertensives do not know they have high blood pressure, 50% of those who do know are inadequately treated by their physicians, and 50% who are appropriately treated fail to follow this treatment by taking the right number of pills. What is the probability that at least 7 of the 10 hypertensives know they have high blood pressure?A national study found that treating people appropriately for high blood pressure reduced their overall mortality by 20%. Treating people adequately for hypertension has been difficult because it is estimated that 50% of hypertensives do not know they have high blood pressure, 50% of those who do know are inadequately treated by their physicians, and 50% who are appropriately treated fail to follow this treatment by taking the right number of pills. If the preceding 50% rates were each reduced to 40% by a massive education program, then what effect would this change have on the overall mortality rate among true hypertensives; that is, would the mortality rate decrease and, if so, what percentage of deaths among hypertensives could be prevented by the education program?The presence of bacteria in a urine sample (bacteriuria) is sometimes associated with symptoms of kidney disease in women. Suppose a determination of bacteriuria has been made over a large population of women at one point in time and 5% of those sampled are positive for bacteriuria. If a sample size of 5 is selected from this population, what is the probability that 1 or more women are positive for bacteriuria?The presence of bacteria in a urine sample (bacteriuria) is sometimes associated with symptoms of kidney disease in women. Suppose a determination of bacteriuria has been made over a large population of women at one point in time and 5% of those sampled are positive for bacteriuria. Suppose 100 women from this population are sampled. What is the probability that 3 or more of them are positive for bacteriuria?The presence of bacteria in a urine sample (bacteriuria) is sometimes associated with symptoms of kidney disease in women. Suppose a determination of bacteriuria has been made over a large population of women at one point in time and 5% of those sampled are positive for bacteriuria. What is the probability distribution of X?One interesting phenomenon of bacteriuria is that there is a turnover; that is, if bacteriuria is measured on the same woman at two different points in time, the results are not necessarily the same. Assume that 20% of all women who are bacteriuric at time 0 are again bacteriuric at time 1 (1 year later), whereas only 4.2% of women who were not bacteriuric at time 0 are bacteriuric at time 1. Let X be the random variable representing the number of bacteriuric events over the two time periods for 1 woman and still assume that the probability that a woman will be positive for bacteriuria at any one exam is 5%. What is the mean of X?One interesting phenomenon of bacteriuria is that there is a turnover; that is, if bacteriuria is measured on the same woman at two different points in time, the results are not necessarily the same. Assume that 20% of all women who are bacteriuric at time 0 are again bacteriuric at time 1 (1 year later), whereas only 4.2% of women who were not bacteriuric at time 0 are bacteriuric at time 1. Let X be the random variable representing the number of bacteriuric events over the two time periods for 1 woman and still assume that the probability that a woman will be positive for bacteriuria at any one exam is 5%. What is the variance of X?Otitis media is a disease that occurs frequently in the first few years of life and is one of the most common reasons for physician visits after the routine checkup. A study was conducted to assess the frequency of otitis media in the general population in the first year of life. Table 4.16 gives the number of infants of 2500 infants who were first seen at birth who remained disease-free by the end of the ith month of life, i = 0, 1, , 12. (Assume no infants have been lost to follow-up.) What is the probability that an infant will have one or more episodes of otitis media by the end of the sixth month of life? The first year of life? Table 4.16 Number of infants (of 2500) who remain disease-free at the end of each month during the first year of lifeOtitis media is a disease that occurs frequently in the first few years of life and is one of the most common reasons for physician visits after the routine checkup. A study was conducted to assess the frequency of otitis media in the general population in the first year of life. Table 4.16 gives the number of infants of 2500 infants who were first seen at birth who remained disease-free by the end of the ith month of life, i = 0, 1, , 12. (Assume no infants have been lost to follow-up.) What is the probability that an infant will have one or more episodes of otitis media by the end of the ninth month of life given that no episodes have been observed by the end of the third month of life? Table 4.16 Number of infants (of 2500) who remain disease-free at the end of each month during the first year of lifeOtitis media is a disease that occurs frequently in the first few years of life and is one of the most common reasons for physician visits after the routine checkup. A study was conducted to assess the frequency of otitis media in the general population in the first year of life. Table 4.16 gives the number of infants of 2500 infants who were first seen at birth who remained disease-free by the end of the ith month of life, i = 0, 1,, 12. (Assume no infants have been lost to follow-up.) Suppose an otitis-prone family is defined as one in which at least three siblings of five develop otitis media in the first 6 months of life. What proportion of five-sibling families is otitis prone if we assume the disease occurs independently for different siblings in a family? Table 4.16 Number of infants (of 2500) who remain disease-free at the end of each month during the first year of lifeOtitis media is a disease that occurs frequently in the first few years of life and is one of the most common reasons for physician visits after the routine checkup. A study was conducted to assess the frequency of otitis media in the general population in the first year of life. Table 4.16 gives the number of infants of 2500 infants who were first seen at birth who remained disease-free by the end of the ith month of life, i = 0, 1, , 12. (Assume no infants have been lost to follow-up.) What is the expected number of otitis-prone families of 100 five-sibling families? Table 4.16 Number of infants (of 2500) who remain disease-free at the end of each month during the first year of lifeAn experiment is designed to test the potency of a drug on 20 rats. Previous animal studies have shown that a 10-mg dose of the drug is lethal 5% of the time within the first 4 hours; of the animals alive at 4 hours, 10% will die in the next 4 hours. What is the probability that 3 or more rats will die in the first 4 hours?An experiment is designed to test the potency of a drug on 20 rats. Previous animal studies have shown that a 10-mg dose of the drug is lethal 5% of the time within the first 4 hours; of the animals alive at 4 hours, 10% will die in the next 4 hours. Suppose 2 rats die in the first 4 hours. What is the probability that 2 or fewer rats will die in the next 4 hours?An experiment is designed to test the potency of a drug on 20 rats. Previous animal studies have shown that a 10-mg dose of the drug is lethal 5% of the time within the first 4 hours; of the animals alive at 4 hours, 10% will die in the next 4 hours. What is the probability that 0 rats will die in the 8-hour period?An experiment is designed to test the potency of a drug on 20 rats. Previous animal studies have shown that a 10-mg dose of the drug is lethal 5% of the time within the first 4 hours; of the animals alive at 4 hours, 10% will die in the next 4 hours. What is the probability that 1 rat will die in the 8-hour period?An experiment is designed to test the potency of a drug on 20 rats. Previous animal studies have shown that a 10-mg dose of the drug is lethal 5% of the time within the first 4 hours; of the animals alive at 4 hours, 10% will die in the next 4 hours. What is the probability that 2 rats will die in the 8-hour period?An experiment is designed to test the potency of a drug on 20 rats. Previous animal studies have shown that a 10-mg dose of the drug is lethal 5% of the time within the first 4 hours; of the animals alive at 4 hours, 10% will die in the next 4 hours. Can you write a general formula for the probability that x rats will die in the 8-hour period? Evaluate this formula for x = 0, 1,, 10. (Hint: Use the BINOMDIST function of Excel.)An important issue in assessing nuclear energy is whether excess disease risks exist in the communities surrounding nuclear-power plants. A study undertaken in the community surrounding Hanford, Washington, looked at the prevalence of selected congenital malformations in the counties surrounding the nuclear-test facility [9]. Suppose 27 cases of Downs syndrome are found and only 19 are expected based on Birth Defects Monitoring Program prevalence estimates in the states of Washington, Idaho, and Oregon. Are there significant excess cases in the area around the nuclear-power plant? Suppose 12 cases of cleft palate are observed, whereas only 7 are expected based on Birth Defects Monitoring Program estimates.49PAn important issue in assessing nuclear energy is whether excess disease risks exist in the communities surrounding nuclear-power plants. A study undertaken in the community surrounding Hanford, Washington, looked at the prevalence of selected congenital malformations in the counties surrounding the nuclear-test facility [9]. Do you feel there is a meaningful excess number of cases of cleft palate in the area surrounding the nuclearpower plant? Explain.51P52PA topic of some interest in the genetic literature over at least the past 30 years has been the study of sexratio data. In particular, one suggested hypothesis is that there are enough families with a preponderance of males (females) that the sexes of successive childbirths are not independent random variables but rather are related to each other. This hypothesis has been extended beyond just successive births, so some authors also consider relationships between offspring two birth orders apart (first and third offspring, second and fourth offspring, etc.). Sex-ratio data from the first 5 births in 51,868 families are given in Data Set SEXRAT.DAT (at www.cengagebrain.com). The format of this file is given in Table 4.18 [10]. What are your conclusions concerning the preceding hypothesis based on your analysis of these data?A study considered risk factors for HIV infection among intravenous drug users [11]. It found that 40% of users who had 100 injections per month (light users) and 55% of users who had 100 injections per month (heavy users) were HIV positive. What is the probability that exactly 3 of 5 light users are HIV positive?A study considered risk factors for HIV infection among intravenous drug users [11]. It found that 40% of users who had 100 injections per month (light users) and 55% of users who had 100 injections per month (heavy users) were HIV positive. What is the probability that at least 3 of 5 light users are HIV positive?A study considered risk factors for HIV infection among intravenous drug users [11]. It found that 40% of users who had 100 injections per month (light users) and 55% of users who had 100 injections per month (heavy users) were HIV positive. Suppose we have a group of 10 light users and 10 heavy users. What is the probability that exactly 3 of the 20 users are HIV positive?A study considered risk factors for HIV infection among intravenous drug users [11]. It found that 40% of users who had 100 injections per month (light users) and 55% of users who had 100 injections per month (heavy users) were HIV positive. What is the probability that at least 4 of the 20 users are HIV positive?A study considered risk factors for HIV infection among intravenous drug users [11]. It found that 40% of users who had 100 injections per month (light users) and 55% of users who had 100 injections per month (heavy users) were HIV positive. Is the distribution of the number of HIV positive among the 20 users binomial? Why or why not?A study [12] of incidence rates of blindness among insulin-dependent diabetics reported that the annual incidence rate of blindness per year was 0.67% among 30- to 39-year-old male insulin-dependent diabetics (IDDM) and 0.74% among 30- to 39-year-old female insulin-dependent diabetics. If a group of 200 IDDM 30- to 39-year-old men is followed, what is the probability that exactly 2 will go blind over a 1-year period?A study [12] of incidence rates of blindness among insulin-dependent diabetics reported that the annual incidence rate of blindness per year was 0.67% among 30- to 39-year-old male insulin-dependent diabetics (IDDM) and 0.74% among 30- to 39-year-old female insulin-dependent diabetics. If a group of 200 IDDM 30- to 39-year-old women is followed, what is the probability that at least 2 will go blind over a 1-year period?A study [12] of incidence rates of blindness among insulin-dependent diabetics reported that the annual incidence rate of blindness per year was 0.67% among 30- to 39-year-old male insulin-dependent diabetics (IDDM) and 0.74% among 30- to 39-year-old female insulin-dependent diabetics. What is the probability that a 30-year-old IDDM male patient will go blind over the next 10 years?A study [12] of incidence rates of blindness among insulin-dependent diabetics reported that the annual incidence rate of blindness per year was 0.67% among 30- to 39-year-old male insulin-dependent diabetics (IDDM) and 0.74% among 30- to 39-year-old female insulin-dependent diabetics. After how many years of follow-up would we expect the cumulative incidence of blindness to be 10% among 30-year-old IDDM females, if the incidence rate remains constant over time?A study [12] of incidence rates of blindness among insulin-dependent diabetics reported that the annual incidence rate of blindness per year was 0.67% among 30- to 39-year-old male insulin-dependent diabetics (IDDM) and 0.74% among 30- to 39-year-old female insulin-dependent diabetics. What does cumulative incidence mean, in words, in the context of this problem?An article was published [13] concerning the incidence of cardiac death attributable to the earthquake in Los Angeles County on January 17, 1994. In the week before the earthquake there were an average of 15.6 cardiac deaths per day in Los Angeles County. On the day of the earthquake, there were 51 cardiac deaths. What is the exact probability of 51 deaths occurring on one day if the cardiac death rate in the previous week continued to hold on the day of the earthquake?An article was published [13] concerning the incidence of cardiac death attributable to the earthquake in Los Angeles County on January 17, 1994. In the week before the earthquake there were an average of 15.6 cardiac deaths per day in Los Angeles County. On the day of the earthquake, there were 51 cardiac deaths. Is the occurrence of 51 deaths unusual? (Hint: Use the same methodology as in Example 4.32.)An article was published [13] concerning the incidence of cardiac death attributable to the earthquake in Los Angeles County on January 17, 1994. In the week before the earthquake there were an average of 15.6 cardiac deaths per day in Los Angeles County. On the day of the earthquake, there were 51 cardiac deaths. What is the maximum number of cardiac deaths that could have occurred on the day of the earthquake to be consistent with the rate of cardiac deaths in the past week? (Hint: Use a cutoff probability of .05 to determine the maximum number.)Some previous studies have shown a relationship between emergency-room admissions per day and level of pollution on a given day. A small local hospital finds that the number of admissions to the emergency ward on a single day ordinarily (unless there is unusually high pollution) follows a Poisson distribution with mean = 2.0 admissions per day. Suppose each admitted person to the emergency ward stays there for exactly 1 day and is then discharged. The hospital is planning a new emergency-room facility. It wants enough beds in the emergency ward so that for at least 95% of normal-pollution days it will not need to turn anyone away. What is the smallest number of beds it should have to satisfy this criterion?Some previous studies have shown a relationship between emergency-room admissions per day and level of pollution on a given day. A small local hospital finds that the number of admissions to the emergency ward on a single day ordinarily (unless there is unusually high pollution) follows a Poisson distribution with mean = 2.0 admissions per day. Suppose each admitted person to the emergency ward stays there for exactly 1 day and is then discharged. The hospital also finds that on high-pollution days the number of admissions is Poisson-distributed with mean = 4.0 admissions per day. Answer Problem 4.67 for high-pollution days.Some previous studies have shown a relationship between emergency-room admissions per day and level of pollution on a given day. A small local hospital finds that the number of admissions to the emergency ward on a single day ordinarily (unless there is unusually high pollution) follows a Poisson distribution with mean = 2.0 admissions per day. Suppose each admitted person to the emergency ward stays there for exactly 1 day and is then discharged. On a random day during the year, what is the probability there will be 4 admissions to the emergency ward, assuming there are 345 normal-pollution days and 20 high-pollution days?Some previous studies have shown a relationship between emergency-room admissions per day and level of pollution on a given day. A small local hospital finds that the number of admissions to the emergency ward on a single day ordinarily (unless there is unusually high pollution) follows a Poisson distribution with mean = 2.0 admissions per day. Suppose each admitted person to the emergency ward stays there for exactly 1 day and is then discharged. Answer Problem 4.67 for a random day during the year. The hospital is planning a new emergency-room facility. It wants enough beds in the emergency ward so that for at least 95% of normal-pollution days it will not need to turn anyone away. What is the smallest number of beds it should have to satisfy this criterion?The number of legal induced abortions per year per 1000 U.S. women ages 1544 [14] is given in Table 4.19. For example, of 1000 women ages 1544 in 1980, 25 had a legal induced abortion during 1980. If we assume (1) no woman has more than 1 abortion and (2) the probability of having an abortion is independent across different years, what is the probability that a 15-yearold woman in 1975 will have an abortion over her 30 years of reproductive life (ages 1544, or 19752004)? Table 4.19 Annual incidence of legal induced abortions by time period Studies have been undertaken to assess the relationship between abortion and the development of breast cancer. In one study among nurses (the Nurses Health Study II), there were 16,359 abortions among 2,169,321 person-years of follow-up for women of reproductive age. (Note: 1 person-year = 1 woman followed for 1 year.)Table 4.19 Annual incidence of legal induced abortions by time period Studies have been undertaken to assess the relationship between abortion and the development of breast cancer. In one study among nurses (the Nurses Health Study II), there were 16,359 abortions among 2,169,321 person-years of follow-up for women of reproductive age. (Note: 1 person-year = 1 woman followed for 1 year.) What is the expected number of abortions among nurses over this time period if the incidence of abortion is 25 per 1000 women per year and no woman has more than 1 abortion?Some previous studies have shown a relationship between emergency-room admissions per day and level of pollution on a given day. A small local hospital finds that the number of admissions to the emergency ward on a single day ordinarily (unless there is unusually high pollution) follows a Poisson distribution with mean = 2.0 admissions per day. Suppose each admitted person to the emergency ward stays there for exactly 1 day and is then discharged. Does the abortion rate among nurses differ significantly from the national experience? Why or why not? (Hint: Use the Poisson distribution.) A yes/no answer is not acceptable.80P81P82PThe two-stage model of carcinogenesis is based on the premise that for a cancer to develop, a normal cell must first undergo a first hit and mutate to become a susceptible or intermediate cell. An intermediate cell then must undergo a second hit and mutate to become a malignant cell. A cancer develops if at least one cell becomes a malignant cell. This model has been applied to the development of breast cancer in females (Moolgavkar et al. [15]). Suppose there are 108 normal breast cells and 0 intermediate or malignant breast cells among 20-year-old females. The probability that a normal breast cell will mutate to become an intermediate cell is 107 per year. What is the probability that there will be at least 5 intermediate cells by age 21? (Hint: Use the Poisson distribution.)The two-stage model of carcinogenesis is based on the premise that for a cancer to develop, a normal cell must first undergo a first hit and mutate to become a susceptible or intermediate cell. An intermediate cell then must undergo a second hit and mutate to become a malignant cell. A cancer develops if at least one cell becomes a malignant cell. This model has been applied to the development of breast cancer in females (Moolgavkar et al. [15]). Suppose there are 108 normal breast cells and 0 intermediate or malignant breast cells among 20-year-old females. The probability that a normal breast cell will mutate to become an intermediate cell is 107 per year. What is the expected number of intermediate cells by age 45? The probability that an intermediate cell will mutate to become a malignant cell is 5 107 per year.The two-stage model of carcinogenesis is based on the premise that for a cancer to develop, a normal cell must first undergo a first hit and mutate to become a susceptible or intermediate cell. An intermediate cell then must undergo a second hit and mutate to become a malignant cell. A cancer develops if at least one cell becomes a malignant cell. This model has been applied to the development of breast cancer in females (Moolgavkar et al. [15]). Suppose there are 108 normal breast cells and 0 intermediate or malignant breast cells among 20-year-old females. The probability that a normal breast cell will mutate to become an intermediate cell is 107 per year. Suppose a woman has 300 intermediate cells by age 45. What is the probability that she develops breast cancer by age 46? By age 50? (Hint: Use the Poisson approximation to the binomial distribution.)The data in Table 4.20 were reported by men in the Health Professionals Follow-up Study on the number of teeth lost over a 1-year period (January 1, 1987, to December 31, 1987). Table 4.20 Distribution of number of teeth lost from January 1, 1987, to December 31, 1987, among 38,905 men in the Health Professionals Follow-up Study If we assume the average number of teeth lost in the 59 group is 7 teeth and the average number of teeth lost in the 10+ group is 12 teeth, what is the best estimate of the average number of teeth lost per year?The data in Table 4.20 were reported by men in the Health Professionals Follow-up Study on the number of teeth lost over a 1-year period (January 1, 1987, to December 31, 1987). Table 4.20 Distribution of number of teeth lost from January 1, 1987, to December 31, 1987, among 38,905 men in the Health Professionals Follow-up Study Suppose that on January 1, 1987, a man is 50 years old, that he will live for 30 more years (until 2016), and that the rate of tooth loss over this 30-year period is the same as in 1987. If a man has 13 teeth remaining on January 1, 1987, what is the probability he will need dentures (have 10 or fewer teeth remaining) during his 30-year lifetime? (Hint: Use the Poisson distribution.)88PSuppose the number of admissions to the emergency room at a small hospital follows a Poisson distribution but the incidence rate changes on different days of the week. On a weekday there are on average two admissions per day, while on a weekend day there is on average one admission per day. What is the probability of at least one admission on a Wednesday?Suppose the number of admissions to the emergency room at a small hospital follows a Poisson distribution but the incidence rate changes on different days of the week. On a weekday there are on average two admissions per day, while on a weekend day there is on average one admission per day. What is the probability of at least one admission on a Saturday?Suppose the number of admissions to the emergency room at a small hospital follows a Poisson distribution but the incidence rate changes on different days of the week. On a weekday there are on average two admissions per day, while on a weekend day there is on average one admission per day. What is the probability of having 0, 1, and 2+ admissions for an entire week, if the results for different days during the week are assumed to be independent?92P93P94PSuppose a city is divided into eight census tracts as shown in Table 4.21. What is the expected number of cases over 1 year in the city? Table 4.21 Relationship between incidence of birth defects and census tractA study was performed concerning medical emergencies on commercial airline flights (Peterson, et al., [16]). A database was constructed based on calls to a medical communications center from 5 domestic and international airlines representing approximately 10% of global passenger flight volume from January 1, 2008 to October 31, 2010. There were 11,920 in-flight medical emergencies (IFM) among 7,198,118 flights during the study period. Assume for this entire problem that there is at most 1 IFM per flight. Suppose a flight attendant works on 2 flights per day for each of 300 days per year. What is the probability that the flight attendant will encounter at least one IFM over a 1-year period? Hint: Use the Poisson approximation to the binomial distribution.Suppose the flight attendants total duration of employment is 20 years. What is the probability that he/she encounters at least 10 IFMs on his/her flights over a 20-year period? Make the same assumptions as in Problem 4.96. Hint: Use a computer program (e.g., Excel, Stata, or R) to solve this problem.The more significant IFMs result in an aircraft being forced to land in an airport other than its original destination due to the medical emergency (referred to as an aircraft diversion or DIV). Suppose that 875 out of the 11,920 IFMs (7.3%) resulted in a DIV. Calculate the probability that a flight attendant encounters at least one DIV over his/her 20-year working lifetime. Make the same assumptions as in Problem 4.96. Hint: Use a computer program (e.g., Excel, Stata, or R) to solve this problem.