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All Textbook Solutions for Introduction to Probability and Statistics

Cholesterol, continued Refer to Exercise 10.16. Since n30, use the methods of Chapter 8 tocreate a large-sample 95% confidence interval for theaverage serum cholesterol level for L.A. Countyemployees. Compare the two intervals. (HINT: Thetwo intervals should be quite similar. This is thereason we choose to approximate the sampledistribution xs/n of with a z-distribution when n30.)Give the number of degrees of freedom for s2, the pooled estimator of 2, in these cases: a. n1=16,n2=8 b. n1=10,n2=12 c. n1=15,n2=3 10.19E 10.20E 10.21E 10.22E The MINITAB printout shows a test for the difference in two population means. MINITAB output for Exercise 10.23 Two-Sample T-Test and Cl: Sample 1, Sample 2 Two-sample T for Sample 1 vs Sample 2 Do the two sample standard deviations indicate that the assumption of a common population variance is reasonable? What is the observed value of the test statistic? What is the p-value associated with this test? What is the pooled estimate s2of the population variance? Use the answers to part b to draw conclusions about the difference in the two population means. Find the 95% confidence interval for the difference in the population means. Does this interval confirm your conclusions in part d?10.24E Healthy Teeth Jan Lindhe conducted a studyon the effect of an oral antiplaque rinse on plaquebuildup on teeth.6Fourteen people whose teeth werethoroughly cleaned and polished were randomlyassigned to two groups of seven subjects each. Bothgroups were assigned to use oral rinses (no brushing)for a 2-week period. Group 1 used a rinse thatcontained an antiplaque agent. Group 2, the controlgroup, received a similar rinse except that, unknown to the subjects, the rinse contained no antiplaque agent. Aplaque index x, a measure of plaque buildup, wasrecorded at 14 days with means and standard deviations for the two groups shown in the table. State the null and alternative hypotheses that shouldbe used to test the effectiveness of the antiplaqueoral rinse. Do the data provide sufficient evidence to indicatethat the oral antiplaque rinse is effective? Test using =.05. Find the approximate p-value for the test.10.26E 10.27E Disinfectants An experiment published in TheAmerican Biology Teacher studied the efficacy of using 95% ethanol or 20% bleach as a disinfectant in removingbacterial and fungal contamination when culturing planttissues. The experiment was repeated 15 times with eachdisinfectant, using eggplant as the plant tissue being cultured.8Five cuttings per plant were placed on a petri dishfor each disinfectant and stored at 25C for 4 weeks. Theobservation reported was the number of uncontaminatedeggplant cuttings after the 4-week storage. Pooled variance 1.47619 Are you willing to assume that the underlying variances are equal? Using the information from part a, are you willingto conclude that there is a significant difference in the mean numbers of uncontaminated eggplants forthe two disinfectants tested?10.29E 10.30E 10.31E 10.32E Freestyle Swimmers, continued Refer toExercise 10.32. Construct a lower 95% one-sided confidence bound for the difference in the average timesfor the two swimmers. Does this interval confirm yourconclusions in Exercise 10.32?10.34E 10.35E 10.36E 10.37E 10.38E 10.39E Runners and Cyclists II Refer to Exercise 10.27. In addition to the compartment pressures, the level of creatine phosphokinase (CPK) in bloodsamples, a measure of muscle damage, was determined for each of 10 runners and 10 cyclists before and afterexercise.7The data summary-CPK values in units/liter-is as follows: Test for a significant difference in mean CPK values for runners and cyclists before exercise under the assumption that 1222; use =.05.Find a95% confidence interval estimate for the corresponding difference in means. Test for a significant difference in mean CPK values for runners and cyclists after exercise under the assumption that 1222; use =.05. Find a 95%confidence interval estimate for the correspondingdifference in means. Test for a significant difference in mean CPKvalues for runners before and after exercise. Find a 95% confidence interval estimate for thedifference in mean CPK values for cyclists beforeand after exercise. Does your estimate indicatethat there is no significant difference in mean CPKlevels for cyclists before and after exercise?10.41E No Left Turn An experiment was conducted to compare the mean reaction times to twotypes of traffic signs: prohibitive (No Left Turn) and permissive (Left Turn Only). Ten drivers were included inthe experiment. Each driver was presented with 40 traffic signs. 20 prohibitive and 20 permissive, in randomorder. The mean time to reaction (in milliseconds) wasrecorded for each driver and ¡s shown here. MS Excel printout for Exercise 10.42 Explain why this is a paired-difference experimentand give reasons why the pairing should be usefulin increasing information on the difference betweenthe mean reaction times to prohibitive and permissive traffic signs. Use the Excel printout to determine whether there isa significant difference in mean reaction times toprohibitive and permissive traffic signs. Use thep-value approach.Healthy Teeth II Exercise 10.25 describes adental experiment conducted to investigate the effectiveness of an oral rinse used to inhibit the growth of plaqueon teeth. Subjects were divided into two groups: Onegroup used a rinse with an antiplaque ingredient, and thecontrol group used a rinse containing inactive ingredients. Suppose that the plaque growth on each person’steeth was measured after using the rinse after 4 hoursand then again after 8 hours. If you wish to estimate thedifference in plaque growth from 4 to 8 hours, shouldyou use a confidence interval based on a paired or anunpaired analysis? Explain.10.44E 10.45E 10.46E 10.47E 10.48E 10.49E 10.50E A random sample of size n=7 from a normalpopulation produced these measurements: 1.4, 3.6, 1.7, 2.0, 3.3, 2.8, 2.9. Calculate the sample variance, s2. Construct a 95% confidence interval for the population variance, 2. Test H0:2=.8 versus Ha:2.8 using =.05.State your conclusions. What is the approximate p-value for the test in part c?10.52E 10.53E 10.54E 10.55E 10.56E 10.57E 10.58E 10.59E 10.60E 10.61E 10.62E 10.63E 10.64E 10.65E 10.66E 11.56SE 11.57SE 11.58SE 11.59SE 11.60SE 11.61SE 11.62SE 11.63SE 11.64SE 11.65SE 11.66SE 11.67SE Yield of Wheat The yields of wheat(in bushels per acre) were compared for five different varieties, A, B, C, D, and E, at six different locations. Each variety was randomly assigned to a plot at each location. The results of the experiment are shown in the accompanying table, along with a MINITAB printout of the analysis of variance. Analyze the experiment using the appropriate method. Identify the treatments or factors of interest to the researcher and investigate any effects that exist. Use the diagnost ic plots to comment on the validity of the analysis of variance assumptions. What are the practical implications of this experiment? Write a paragraph explaining the results of your analysis 11.69SE Professor’s Salaries In a study of starting salaries of assistant professors. five male and five female beginning assistant professors at each of two types of institutions granting doctoral degrees were polled and their initial starting salaries were recorded. The results of the survey in thousands of dollars are given in the following table. a. What type of design was used in collecting these data? b. Use an analysis of variance to test if there are significant differences in gender, in type of institution, and to test for a significant interaction of gender type of institution. c. Find a 95% confidence interval estimate for the difference in starting salaries for male assistant professors and female assistant professors. Interpret this interval in terms of a gender difference in starting salaries. d. Summarize the results of your analysis.11.71SE 11.72SE 11.73SE 11.74SE 1CS 2CS 3CS 4CS 11.1E 11.2E 11.3E 11.4E 11.5E 11.6E These data are observations collected using a completely randomized design: Calculate CM and Total SS. Calculate SST and MST. Calculate SSE and MSE. Construct an ANOVA table for the data. State the null and alternative hypotheses for an analysis of variance F-test. Use the p-value approach to determine whether there is a difference in the three population means.11.8E 11.9E Reducing Hostility A clinical psychologist wished to compare three methods for reducing hostility levels in university students using a certain psychological test (HLT). High scores on this test were taken to indicate great hostility, and 11 students who got high and nearly equal scores were used in the experiment. Five were selected at random from among the 11 students and treated by method A. three were taken at random from the remaining six students and treated by method 13, and the other three students were treated by method C. All treatments continued throughout a semester, when the HLT test was given again. The results are shown in the table. Perform an analysis of variance for this experiment. Do the data provide sufficient evidence to indicate a difference in mean student scores after treatment for the three methods?11.11E Assembling Electronic Equipment An experiment was conducted to compare the effectiveness of three training programs. A, B, and C, in training assemblers of a piece of electronic equipment. Fifteen employees were randomly assigned, five each, to the three programs. After completion of the program, each person was required to assemble tour pieces of the equipment, and the average length of time required to complete the assembly was recorded. Several of the employees resigned during the course of the program; the remainder were evaluated, producing the data shown in the accompanying table. Use the Excel printout to answer the questions. Excel printout to answer the questions. a. Do the data provide sufficient evidence to indicate a difference in mean assembly times for people trained by the three programs? Give the p—value for the test and interpret its value. b. Find a 99% confidence interval for the difference in mean assembly times between persons trained by programs A and B. c. Find a 99% confidence interval for the mean assembly times for persons trained by program A. d. Do you think the data will satisfy (approximately) the assumption that they have been selected from normal populations? Why? MS Excel output for Exercise 11.12 11.13E 11.14E 11.15E 11.16E The Cost of Lumber A national home builder wants to compare the prices per 1000 board feet of standard or better grade green Douglas fir framing lumber. He randomly selects five suppliers in each of the four states where the builder is planning to begin construction. The prices are given in the table What type of experimental design has been used? Construct the analysis of variance table for this data. Do the data provide sufficient evidence to indicate that the average price per 1000 board feet of Douglas fir differs among the four states? Test using =0.5.11.18E 11.19E 11.20E 11.21E 11.22E 11.23E 11.24E 11.25E 11.26E 11.27E 11.28E 11.29E Do the data of Exercise 11.28 provide sufficient evidence to indicate difference among the treatmen means? Test usign =0.5.11.31E 11.32E 11.34E The partially completed ANOVA table for a randomized block design is presented here: How many blocks are involved in the design? How many observations are in each treatment total? How many observations are in each block total? Fill in the blanks in the ANOVA table. Do the data present sufficient evidence to indicate difference among the treatment means? Test using =0.5. Do the data present sufficient evidence to indicate difference among the block means? Test using =0.5.Gas Mileage A study was conducted to compare automobile gasoline mileage for three formulations of gasoline. Four automobiles, all of the same make and model, were used in the experiment. and each bormulation was tested in each automobile. Using each formulation in the same automobile has the effect of eliminating (blocking out) automobile—to—automobile variability. The data (in miles per gallon) follow. Do the data provide sufficient evidence to indicate a difference in mean mileage per gallon for the three gasoline formulations? Is there evidence of a difference in mean mileage for the four automobiles? Suppose that prior to looking at the data, you had decided to compare the mean mileage per gallon for formulations A and B. Find a 90% confidence interval for this difference. Use an appropriate method to identify the pairwise differences, if any, in the average mileages for the three formulations.11.38E 11.39E Digitalis and Calcium Uptake A study was conducted to compare the effects of three levels of digitalis on the levels of calcium in the heart muscles of dogs. Because general level of calcium uptake vanes from one animal to another, the tissue for a heart muscle was regarded as a block, and comparisons of the three digitalis levels (treatments) were made within a given animal. The calcium uptakes for the three levels of digitalis, A, B, and C, were compared based on the heart muscles of four dogs and the results are given in the table. Use the Excel printout to answer the questions. a. How many degrees of freedom are associated with SSE’? b. Do the data present sufficient evidence to indicate a difference in the mean uptakes of calcium for the three levels of digitalis? c. Use Tukey’s method of paired comparisons with =0.1 to rank the mean calcium uptakes for the three levels of digitalis. d. Do the data indicate a difference in the mean uptakes of calcium for the tour heart muscles’? e. Use Tukey’s method of paired comparisons with =0.1 to rank the mean calcium uptakes for the heart muscles of the four dogs used in the experiment. Are these results of any practical value to the researcher’? f. Give the standard error of the difference between the mean calcium uptakes for two level of digitalis. g. Find a 95% confidence interval for the difference in mean responses between treatments A and B.Bidding on Construction Jobs A building contractor employs three construction engineers. A, B, and C, to estimate and bid on jobs. To determine whether one tends to be a more conservative (or liberal) estimator than the others, the contractor selects four projected construction jobs and has each estimator independently estimate the cost (in dollars per square foot) of each job. The data are shown in the table: Analyze the experiment using the appropriate methods. Identity the blocks and treatments, and investigate any possible differences in treatment means. If any differences exist, use an appropriate method to specifically identity where the differences lie. Has blocking been effective in this experiment? What are the practical implications of the experiment’? Present your results in the form of a report.Premium Equity? The cost of auto insurance varies by coverage, location, and the driving record of the driver. The following are estimates of the annual cost for standard coverage as of January 1, 2011 for a male driver with 6—8 years of experience, driving a Honda Accord with no accidents or violations.3 (These are quotes and not premiums a. What type of design was used in collecting these data? b. Is there sufficient evidence to indicate that insurance premiums for the same type of coverage differs from company to company’? c. Is there sufficient evidence to indicate that insurance premiums vary from location to location’? d. Use Tukey’s procedure to determine which insurance companies listed here differ from others in the premiums they charge for this typical client. Use =.05. e. Summarize your bindings.11.43E 11.44E 11.45E 11.46E 11.47E 11.48E 11.49E Demand for Diamonds A chain of jewelry stores conducted an experiment to investigate the effect of price markup and location on the demand for its diamonds. Six small—town stores were selected for the study, as well as six stores located in large suburban malls. Two stores in each of these locations were assigned to each of three item percentage markups. The percentage gain (or loss) in sales for each store was recorded at the end of 1 month. The data are shown in the accompanying table. a. Do the data provide sufficient evidence to indicate an interaction between markup and location? Test using =.05. b. What are the practical implications of your test in part a? c. Draw a line graph similar to Figure 11 .11 to help visualize the results of this experiment. Summarize the results. d. Find a 95% confidence interval for the difference in mean change in sales for stores in small towns versus those in suburban malls if the stores are using price markup 3.Terrain Visualization A study was conducted to determine the effect of two lactors on terrain visualization training for soldiers.4 During the raining programs. participants viewed contour maps of various terrains and then were permitted to view a computer reconstruction of the terrain as it would appear from a specified angle. The two factors investig ated in the experiment were the participants’ spatial abilities (abilities to visualize in three dimensions) and the viewing procedures (active or passive). Active participat ion permitted participants to view the computer— generated reconstructions of the terrain from any and all angles. Passive participation gave the participants a set of preselected reconstructions of the terrain. Participants were tested according to spatial ability, and from the test scores 20 were categorized as possessing high spatialability. 20 medium. and 20 low. Then 10 participants within each of these groups were assigned to each of the two training modes, active or passive. The accompanying tables are the ANOVA table computed by the researchers and the table of the treatment means. a. Explain how the authors arrived at the degrees of freedom shown in the ANOVA table. b. Are the F-values correct? c. Interpret the test results. What are their practical implications? d. Use Table 6 in Appendix I to approximate the p-values for the F statistics shown in the ANOVA table. Source: H.F. Barsam and Z.M. Simutis. “Computer-Based Graphics for Terrain Visualization Training. Human Factors, no. 26, 1984. Copyright 1984 by the Human Factors Society. Inc. Reproduced by permission.11.52E 11.53E 11.54E 11.55E 12.61SE 12.62SE 12.63SE 12.64SE 12.65SE 12.66SE 12.67SE Tennis, Anyone? If you play tennis, you know that tennis racquets vary in their physical characteristics. The data in the accompanying table give measures of bending stiffness and twisting stiffness as measured by engineering tests for 12 tennis racquets: a. If a racquet has bending stiffness, is it also likely to have twisting stiffness’? Do the data provide evidence that x and y are positively correlated? b. Calculate the coefficient of determination r2 andinterpret its value.12.69SE 12.70SE 12.71SE Movie Reviews How many weeks cana movie run and still make a reasonable profit? The data that follow show the number of weeks inrelease (x) and the gross to date (y) for the top12 movies during a recent week.17 Plot the points in a scatterplot. Does it appear that the relationship between x and y is linear? Howwould you describe the direction and strength of the relationship? Calculate the value of r2. What percentage of the overall variation is explained by using the linear model rather than Yto predict the response variable y? What is the regression equation? Do the data provide evidence to indicate that x and y are linearlyrelated? Test using a 5% significance level. Given the results of parts b and c, is it appropriate to use the regression line for estimation and prediction? Explain your answer.In addition to increasingly large bounds onerror, why should an experimenter refrain from predicting y for values of x outside the experimental region?12.74SE 12.75SE 12.76SE 12.77SE 12.78SE 1CS 2CS 3CS 12.1E 12.2E 12.3E 12.4E 12.5E You are given five points with these coordinates: Use the data entry method on your scientific orgraphing calculator to enter the n=5observations.Find the sums of squares and cross-products, Sxx, Sxy, and Syy. Find the least-squares line for the data. Plot the five points and graph the line in part b.Does the line appear to provide a good fit to thedata points? Construct the ANOVA table for the linear regression.12.7E 12.8E 12.9E 12.10E 12.11E 12.12E 12.13E 12.14E 12.15E 12.16E 12.17E 12.18E 12.19E 12.20E 12.21E 12.22E 12.23E 12.24E Professor Asimov, continued Refer to thedata in Exercise 12.9, relating x, the number of bookswritten by Professor Isaac Asimov, to y, the number of months he took to write his books (in increments of 100). The data are reproduced below. Do the data support the hypothesis that =0?Use the p-value approach, bounding the p-value using Table 4 of Appendix I. Explain your conclusions inpractical terms. Use the ANOVA table in Exercise 12.9, part c, to calculate the coefficient of determination r2. What percentage reduction in the total variationis achieved by using the linear regressionmodel? Plot the data or refer to the plot in Exercise 12.9, part b. Do the results of parts a and b indicate thatthe model provides a good fit for the data? Are there any assumptions that may have been violated in fitting the linear model?12.26E 12.27E 12.28E 12.29E 12.30E 12.31E 12.32E 12.33E 12.34E 12.35E 12.36E 12.37E 12.38E Refer to Exercise 12.7. Portions of the MINITAB printout are shown here. MINITAB Output for Exercise 12.40 Regression Analysis: y versus x Find a 95% confidence interval for the average value of y when x=2. Find a 95% prediction interval for some value of ‘‘to be observed in the future when x=2. The last line in the third section of the printoutindicates a problem with one of the fitted values. What value of x corresponds to the fitted value y=1.5429? What problem has the MINITAB program detected?12.41E 12.42E 12.43E 12.44E 12.45E 12.46E 12.47E 12.48E 12.49E 12.50E 12.51E 12.52E 12.53E 12.54E 12.55E 12.56E 12.57E 12.58E 12.59E Baseball Stats Does a team’s batting average depend in any way on the number of home runs hit by the team? The data in the table showthe number of team home runs and the overall team batting average for eight selected major league teams for the 2010 season.14 Plot the points using a scatterplot. Does it appear that there is any relationship between total homeruns and team batting average? Is there a significant positive correlation between total home runs and team batting average? Test atthe 5% level of significance. Do you think that the relationship between thesetwo variables would be different if we had lookedat the entire set of major league franchises?13.25SE 13.26SE 13.28SE 13.29SE 13.30SE 13.31SE 13.32SE 13.33SE Quality Control A manufacturer recorded the number of defective items(y) produced on a given day by each of 10 machine operators and also recorded the average output per hour ( x1 )for each operator and the time in weeks from the last machine service ( x2 ). The printout that follows resulted when these data were analyzed using the MINITAB package using the model: E(y)=0+1x1+2x2 a. Interpret R2 and comment on the fit of the model. b. Is there evidence to indicate that the model contributes significantly to the prediction of y at the =.01 level of significance? c. What is the prediction equation relating y and x1 when x2=4? d. Use the fitted prediction equation to predict the number of defective items produced for an operator whose average output per hour is 25 and whose machine was serviced 3 weeks ago. e. What do the residual plots tell you about the validity of the regression assumptions?13.35SE 13.36SE 13.1E 13.2E Suppose that you fit the model E(y)=0+1x1+2x2+3x3 to 15 data points and found F equal to 57.44. Do the data provide sufficient evidence to indicate that the model contributes information for the prediction of y? Test usign a 5% level of significance. Use the value of F to calculate R2. Interpret its value.13.4E 13.5E 13.6E 13.7E 13.8E 13.9E College Textbooks A publisher of college textbooks conducgted a study to relate profit per text y to cost of sales x over a 6-year period when its sales force (and sales costs) were growing raidly. These inflation-adjusted data (in thousands of dollars) were collect: Expecting profit per book to rise and then plateau, the publisher fitted the model E(y)=0+1x+2x2 to the data. Plot the data points. Does it look as though the quadratic model is necessary? Find s on the printout. Confirm that s=SSEnk1 Do the data provide sufficient evidence to indicate that the model contributes information for the pred iction of? What is the p-value for this test. and what does it mean? What sign would you expect the actual value of 2 to have? Find the value of 2 in the printout. Does this value confirm your expecLation? Do the data indicate a significant curvature in the relationship between y and x? Test at the 5% level of significance. What conclusions can you draw from the accompaflying residual plots?13.11E 13.12E 13.13E 13.14E 13.15E 13.16E 13.17E 13.18E 13.19E 13.20E 13.21E 13.22E 13.23E Construction Projects In a study to examine the relationship between the time required to complete a construction project and several pertinent independent variables, an analyst compiled a list of four variables that might be useful in predicting the time to completion. These four variables were size of the contract, x1(in1000unit), number of workd ays adversely affected by the weather x2 , number of subcontractors involved in the project x4 , and a varia ble x3 that measured the presence (x3=1) or absence (x3=0) of a workers’ strike during the construction. Fifteen construction projects were randomly chosen, and each of the four variables as well as the time to completion were measured. An analysis of these data using a first-order model in x1,x2,x3 and x4 produced the following printout. Give a complete analysis of the printout and interpret your results. What can you say about the apparent contribution of x1 and x2 in predicting y?14.35SE 14.36SE 14.37SE 14.38SE 14.39SE 14.40SE 14.41SE 14.42SE 14.43SE 14.44SE 14.45SE 14.46SE 14.47SE 14.48SE 14.49SE 14.50SE 14.51SE 14.52SE 14.53SE 14.54SE 14.55SE 14.56SE 14.57SE 14.58SE 14.59SE 14.60SE 14.61SE 1CS 2CS 3CS 4CS 5CS 14.1E 14.2E 14.3E 14.4E Suppose that a response can fall into one of k=5 categories with probabilities p1,p2,.....,p5 and that n=300 responses produced these category counts: a. Are the five categories equally likely to occur? How would you test this hypothesis? b. If you were to test this hypothesis using the chis quare statistic. how many degrees of freedom would the test have? c. Find the critical value of x2 that defines the rejection region with =.05 . d. Calculate the observed value of the test statistic. e. Conduct the test and state your conclusions.14.6E 14.7E 14.8E 14.9E Mortality Statistics Medical statistics show that deaths due to four major diseases—call them A, B, C, and D—account for 15%, 21%, 18%, and 14%, respectively, of all nonaccidental deaths. A study of the causes of 308 nonaccidental deaths at a hospital gave the following counts: Do these data provide sufficient evidence to indicate that the proportions of people dying of diseases A, B, C, and D at this hospital differ from the proportions accumulated for the population at large?14.11E 14.12E 14.13E 14.14E 14.15E 14.16E 14.17E 14.18E 14.19E 14.20E 14.21E 14.22E Hair Color The hair and eye color that follows was self-reported by a sample of Caucasian Americans born between 1957 and 1965 (currently 45—53 years old).6 The following data was adapted from that study. Is there sufficient evidence to conclude that the proportion of individuals with these hair colours differ for males and females? Use =.05 . Are there any cells with an expected number less than five? If so, combine those cells with those next to it and reanalyze the data. Do the end results differ?14.24E 14.25E 14.26E 14.27E

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