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All Textbook Solutions for Understandable Statistics: Concepts and Methods
Critical Thinking Consider a test for . If the P-value is such that you can reject H0 for = 0.01, can you always reject H0 for = 0.05? Explain.Critical Thinking If sample data is such that for a one-tailed test of you can reject H0 at the 1% level of significance, can you always reject H0 for a two-tailed test at the same level of significance? Explain.Basic Computation: P-value Corresponding to t Value For a Students t distribution with d.f. = 10 and t = 2.930, (a) find an interval containing the corresponding P-value for a two-tailed test. (b) find an interval containing the corresponding P-value for a right-tailed test.Basic Computation: P-value Corresponding to t Value For a Students t distribution with d.f. = 16 and t = 1.830, (a) find an interval containing the corresponding P-value for a two-tailed test. (b) find an interval containing the corresponding P-value for a left-tailed test.Basic Computation: Testing , Unknown A random sample of 25 values is drawn from a mound-shaped and symmetric distribution. The sample mean is 10 and the sample standard deviation is 2. Use a level of significance of 0.05 to conduct a two-tailed test of the claim that the population mean is 9.5. (a) Check Requirements Is it appropriate to use a Students t distribution? Explain. How many degrees of freedom do we use? (b) What are the hypotheses? (c) Compute the t value of the sample test statistic. (d) Estimate the P-value for the test. (e) Do we reject or fail to reject H0? (f) Interpret the results.Basic Computation: Testing , Unknown A random sample has 49 values. The sample mean is 8.5 and the sample standard deviation is 1.5. Use a level of significance of 0.01 to conduct a left-tailed test of the claim that the population mean is 9.2. (a) Check Requirements Is it appropriate to use a Students t distribution? Explain. How many degrees of freedom do we use? (b) What are the hypotheses? (c) Compute the t value of the sample test statistic. (d) Estimate the P-value for the test. (e) Do we reject or fail to reject H0? (f) Interpret the results.Please provide the following information for Problems 1122. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the appropriate sampling distribution value of the sample test statistic. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not given in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more conservative answer. 11. Meteorology: Storms Weatherwise is a magazine published by the American Meteorological Society. One issue gives a rating system used to classify Noreaster storms that frequently hit New England and can cause much damage near the ocean. A severe storm has an average peak wave height of = 16.4 feet for waves hitting the shore. Suppose that a Noreaster is in progress at the severe storm class rating. Peak wave heights are usually measured from land (using binoculars) off fixed cement piers. Suppose that a reading of 36 waves showed an average wave height of x=17.3 feet. Previous studies of severe storms indicate that = 3.5 feet. Does this information suggest that the storm is (perhaps temporarily) increasing above the severe rating? Use = 0.01.Please provide the following information for Problems 1122. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the appropriate sampling distribution value of the sample test statistic. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not given in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more conservative answer. 12. Medical: Blood Plasma Let x be a random variable that represents the pH of arterial plasma (i.e., acidity of the blood). For healthy adults, the mean of the x distribution is = 7.4 (Reference: The Merck Manual, a commonly used reference in medical schools and nursing programs). A new drug for arthritis has been developed. However, it is thought that this drug may change blood pH. A random sample of 31 patients with arthritis took the drug for 3 months. Blood tests showed that x=8.1 with sample standard deviation s = 1.9. Use a 5% level of significance to test the claim that the drug has changed (either way) the mean pH level of the blood.Please provide the following information for Problems 1122. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the appropriate sampling distribution value of the sample test statistic. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not given in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more conservative answer. 13. Wildlife: Coyotes A random sample of 46 adult coyotes in a region of northern Minnesota showed the average age to be x=2.05 years, with sample standard deviation s = 0.82 years (based on information from the book Coyotes: Biology, Behavior and Management by M. Bekoff, Academic Press). However, it is thought that the overall population mean age of coyotes is = 1.75. Do the sample data indicate that coyotes in this region of northern Minnesota tend to live longer than the average of 1.75 years? Use = 0.01.Please provide the following information for Problems 1122. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the appropriate sampling distribution value of the sample test statistic. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not given in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more conservative answer. 14. Fishing: Trout Pyramid Lake is on the Paiute Indian Reservation in Nevada. The lake is famous for cutthroat trout. Suppose a friend tells you that the average length of trout caught in Pyramid Lake is = 19 inches. However, the Creel Survey (published by the Pyramid Lake Paiute Tribe Fisheries Association) reported that of a random sample of 51 fish caught, the mean length was x=18.5 inches, with estimated standard deviation s = 3.2 inches. Do these data indicate that the average length of a trout caught in Pyramid Lake is less than = 19 inches? Use = 0.05.Please provide the following information for Problems 1122. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the appropriate sampling distribution value of the sample test statistic. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not given in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more conservative answer. 15. Investing: Stocks Socially conscious investors screen out stocks of alcohol and tobacco makers, firms with poor environmental records, and companies with poor labor practices. Some examples of good, socially conscious companies are Johnson and Johnson, Dell Computers, Bank of America, and Home Depot. The question is, are such stocks overpriced? One measure of value is the P/E, or price-to-earnings, ratio. High P/E ratios may indicate a stock is overpriced. For the SP stock index of all major stocks, the mean P/E ratio is = 19.4. A random sample of 36 socially conscious stocks gave a P/E ratio sample mean of x=17.9, with sample standard deviation s 5 5.2 (Reference: Morningstar, a financial analysis company in Chicago). Does this indicate that the mean P/E ratio of all socially conscious stocks is different (either way) from the mean P/E ratio of the SP stock index? Use = 0.05.16PPlease provide the following information for Problems 1122. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the appropriate sampling distribution value of the sample test statistic. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not given in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more conservative answer. 17. Medical: Red Blood Cell Count Let x be a random variable that represents red blood cell (RBC) count in millions of cells per cubic millimeter of whole blood. Then x has a distribution that is approximately normal. For the population of healthy female adults, the mean of the x distribution is about 4.8 (based on information from Diagnostic Tests with Nursing Implications, Springhouse Corporation). Suppose that a female patient has taken six laboratory blood tests over the past several months and that the RBC count data sent to the patients doctor are 4.9 4.2 4.5 4.1 4.4 4.3 i. Use a calculator with sample mean and sample standard deviation keys to verify that x=4.40 and s 0.28. ii. Do the given data indicate that the population mean RBC count for this patient is lower than 4.8? Use = 0.05.18PPlease provide the following information for Problems 1122. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the appropriate sampling distribution value of the sample test statistic. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not given in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more conservative answer. 19. Ski Patrol: Avalanches Snow avalanches can be a real problem for travelers in the western United States and Canada. A very common type of avalanche is called the slab avalanche. These have been studied extensively by David McClung, a professor of civil engineering at the University of British Columbia. Slab avalanches studied in Canada have an average thickness of = 67 cm (Source: Avalanche Handbook by D. McClung and P. Schaerer). The ski patrol at Vail, Colorado, is studying slab avalanches in its region. A random sample of avalanches in spring gave the following thicknesses (in cm): i. Use a calculator with mean and standard deviation keys to verify that x61.8 and s 10.6 cm. ii. Assume the slab thickness has an approximately normal distribution. Use a 1% level of significance to test the claim that the mean slab thickness in the Vail region is different from that in Canada.20P21P22PCritical Thinking: One-Tailed versus Two-Tailed Tests (a) For the same data and null hypothesis, is the P-value of a one-tailed test (right or left) larger or smaller than that of a two-tailed test? Explain. (b) For the same data, null hypothesis, and level of significance, is it possible that a one-tailed test results in the conclusion to reject H0 while a two-tailed test results in the conclusion to fail to reject H0? Explain. (c) For the same data, null hypothesis, and level of significance, if the conclusion is to reject H0 based on a two-tailed test, do you also reject H0 based on a one-tailed test? Explain. (d) If a report states that certain data were used to reject a given hypothesis, would it be a good idea to know what type of test (one-tailed or two-tailed) was used? Explain.24P25P26P27P28P29P30PStatistical Literacy To use the normal distribution to test a proportion p, the conditions np 5 and nq 5 must be satisfied. Does the value of p come from H0, or is it estimated by using p from the sample?Statistical Literacy Consider a binomial experiment with n trials and r successes. For a test for a proportion p, what is the formula for the z value of the sample test statistic? Describe each symbol used in the formula.3PCritical Thinking An article in a newspaper states that the proportion of traffic accidents involving road rage is higher this year than it was last year, when it was 15%. Reconstruct the information of the study in terms of a hypothesis test. Discuss possible hypotheses, possible issues about the sample, possible levels of significance, and the absolute truth of the conclusion.Basic Computation: Testing p A random sample of 30 binomials trials resulted in 12 successes. Test the claim that the population proportion of successes does not equal 0.50. Use a level of significance of 0.05. (a) Check Requirements Can a normal distribution be used for the p distribution? Explain. (b) State the hypotheses. (c) Compute p and the corresponding standardized sample test statistic. (d) Find the P-value of the test statistic. (e) Do you reject or fail to reject H0? Explain. (f) Interpretation What do the results tell you?Basic Computation: Testing p A random sample of 60 binomials trials resulted in 18 successes. Test the claim that the population proportion of successes exceeds 18%. Use a level of significance of 0.01. (a) Check Requirements Can a normal distribution be used for the p distribution? Explain. (b) State the hypotheses. (c) Compute p and the corresponding standardized sample test statistic. (d) Find the P-value of the test statistic. (e) Do you reject or fail to reject H0? Explain. (f) Interpretation What do the results tell you?7P8P9PFor Problems 721, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? Do you think the sample size is sufficiently large? Explain. Compute the value of the sample test statistic and corresponding z value. (c) Find the P-value of the test statistic. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. 10. College Athletics: Graduation Rate Women athletes at the University of Colorado, Boulder, have a long-term graduation rate of 67% (Source: Chronicle of Higher Education). Over the past several years, a random sample of 38 women athletes at the school showed that 21 eventually graduated. Does this indicate that the population proportion of women athletes who graduate from the University of Colorado, Boulder, is now less than 67%? Use a 5% level of significance.11PFor Problems 721, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? Do you think the sample size is sufficiently large? Explain. Compute the value of the sample test statistic and corresponding z value. (c) Find the P-value of the test statistic. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. 12. Preference: Color What is your favorite color? A large survey of countries, including the United States, China, Russia, France, Turkey, Kenya, and others, indicated that most people prefer the color blue. In fact, about 24% of the population claim blue as their favorite color (Reference: Study by J. Bunge and A. Freeman-Gallant, Statistics Center, Cornell University). Suppose a random sample of n = 56 college students were surveyed and r = 12 of them said that blue is their favorite color. Does this information imply that the color preference of all college students is different (either way) from that of the general population? Use = 0.05.For Problems 721, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? Do you think the sample size is sufficiently large? Explain. Compute the value of the sample test statistic and corresponding z value. (c) Find the P-value of the test statistic. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. 13. Wildlife: Wolves The following is based on information from The Wolf in the Southwest: The Making of an Endangered Species by David E. Brown (University of Arizona Press). Before 1918, the proportion of female wolves in the general population of all southwestern wolves was about 50%. However, after 1918, southwestern cattle ranchers began a widespread effort to destroy wolves. In a recent sample of 34 wolves, there were only 10 females. One theory is that male wolves tend to return sooner than females to their old territories where their predecessors were exterminated. Do these data indicate that the population proportion of female wolves is now less than 50% in the region? Use = 0.01.14PFor Problems 721, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? Do you think the sample size is sufficiently large? Explain. Compute the value of the sample test statistic and corresponding z value. (c) Find the P-value of the test statistic. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Platos Republic: Syllable Patterns Prose rhythm is characterized by the occurrence of five-syllable sequences in long passages of text. This characterization may be used to assess the similarity among passages of text and sometimes the identity of authors. The following information is based on an article by D. Wishart and S. V. Leach appearing in Computer Studies of the Humanities and Verbal Behavior (Vol. 3, pp. 9099). Syllables were categorized as long or short. On analyzing Platos Republic, Wishart and Leach found that about 26.1% of the five-syllable sequences are of the type in which two are short and three are long. Suppose that Greek archaeologists have found an ancient manuscript dating back to Platos time (about 427347 B.C.). A random sample of 317 five-syllable sequences from the newly discovered manuscript showed that 61 are of the type two short and three long. Do the data indicate that the population proportion of this type of five-syllable sequence is different (either way) from the text of Platos Republic? Use = 0.01.16PFor Problems 721, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? Do you think the sample size is sufficiently large? Explain. Compute the value of the sample test statistic and corresponding z value. (c) Find the P-value of the test statistic. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. 17. Consumers: Product Loyalty USA Today reported that about 47% of the general consumer population in the United States is loyal to the automobile manufacturer of their choice. Suppose Chevrolet did a study of a random sample of 1006 Chevrolet owners and found that 490 said they would buy another Chevrolet. Does this indicate that the population proportion of consumers loyal to Chevrolet is more than 47%? Use = 0.01.18P19P20P21P22PCritical Region Method: Testing Proportions Solve Problem 11 using the critical region method of testing. Hint: See Problem 22. Compare your conclusions with the conclusions obtained by using the P-value method. Are they the same? 22. Critical Region Method: Testing Proportions Solve Problem 9 using the critical region method of testing. Since the sampling distribution of p is the normal distribution, you can use critical values from the standard normal distribution as shown in Figure 8-8 or part (c) of Table 5, Appendix II. Compare your conclusions with the conclusions obtained by using the P-value method. Are they the same? 11. Highway Accidents: DUI The U.S. Department of Transportation, National Highway Traffic Safety Administration, reported that 77% of all fatally injured automobile drivers were intoxicated. A random sample of 27 records of automobile driver fatalities in Kit Carson County, Colorado, showed that 15 involved an intoxicated driver. Do these data indicate that the population proportion of driver fatalities related to alcohol is less than 77% in Kit Carson County? Use = 0.01.24P1P2P3P4P5P6P7PBasic Computation: Paired Differences Test For a random sample of 20 data pairs, the sample mean of the differences was 2. The sample standard deviation of the differences was 5. Assume that the distribution of the differences is mound-shaped and symmetric. At the 1% level of significance, test the claim that the population mean of the differences is positive. (a) Check Requirements Is it appropriate to use a Students t distribution for the sample test statistic? Explain. What degrees of freedom are used? (b) State the hypotheses. (c) Compute the sample test statistic and corresponding t value. (d) Estimate the P-value of the sample test statistic. (e) Do we reject or fail to reject the null hypothesis? Explain. (f) Interpretation What do your results tell you?9P10PFor Problems 921 assume that the distribution of differences d is moundshaped and symmetric. Please provide the following information for Problems 921. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the value of the sample test statistic and corresponding t value. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level a? (e) Interpret your conclusion in the context of the application. In these problems, assume that the distribution of differences is approximately normal. Note: For degrees of freedom d.f. not in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more conservative answer. 11. Ecology: Rocky Mountain National Park The following is based on information taken from Winter Wind Studies in Rocky Mountain National Park by D. E. Glidden (Rocky Mountain Nature Association). At five weather stations on Trail Ridge Road in Rocky Mountain National Park, the peak wind gusts (in miles per hour) for January and April are recorded below. Does this information indicate that the peak wind gusts are higher in January than in April? Use = 0.01.For Problems 921 assume that the distribution of differences d is moundshaped and symmetric. Please provide the following information for Problems 921. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the value of the sample test statistic and corresponding t value. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level a? (e) Interpret your conclusion in the context of the application. In these problems, assume that the distribution of differences is approximately normal. Note: For degrees of freedom d.f. not in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more conservative answer. 12. Wildlife: Highways The western United States has a number of four-lane interstate highways that cut through long tracts of wilderness. To prevent car accidents with wild animals, the highways are bordered on both sides with 12-foot-high woven wire fences. Although the fences prevent accidents, they also disturb the winter migration pattern of many animals. To compensate for this disturbance, the highways have frequent wilderness underpasses designed for exclusive use by deer, elk, and other animals. In Colorado, there is a large group of deer that spend their summer months in a region on one side of a highway and survive the winter months in a lower region on the other side. To determine if the highway has disturbed deer migration to the winter feeding area, the following data were gathered on a random sample of 10 wilderness districts in the winter feeding area. Row B represents the average January deer count for a 5-year period before the highway was built, and row A represents the average January deer count for a 5-year period after the highway was built. The highway department claims that the January population has not changed. Test this claim against the claim that the January population has dropped. Use a 5% level of significance. Units used in the table are hundreds of deer.13P14P15P16PFor Problems 921 assume that the distribution of differences d is moundshaped and symmetric. Please provide the following information for Problems 921. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the value of the sample test statistic and corresponding t value. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level a? (e) Interpret your conclusion in the context of the application. In these problems, assume that the distribution of differences is approximately normal. Note: For degrees of freedom d.f. not in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more conservative answer. 17. Economics: Cost of Living Index In the following data pairs, A represents the cost of living index for housing and B represents the cost of living index for groceries. The data are paired by metropolitan areas in the United States. A random sample of 36 metropolitan areas gave the following information (Reference: Statistical Abstract of the United States, 121st edition). i. Let d be the random variable d = A - B. Use a calculator to verify that d 2.472 and sd 12.124. ii. Do the data indicate that the U.S. population mean cost of living index for housing is higher than that for groceries in these areas? Use = 0.05.18P19P20P21PExpand Your Knowledge: Confidence Intervals for d Using techniques from Section 7.2, we can find a confidence interval for d. Consider a random sample of n matched data pairs A, B. Let d = B A be a random variable representing the difference between the values in a matched data pair. Compute the sample mean d of the differences and the sample standard deviation sd. If d has a normal distribution or is mound-shaped, or if n 30, then a confidence interval for d is dEdd+E where E=tcSdn c = confidence level (0 c 1) tc = critical value for confidence level c and d.f. = n 1 (a) Using the data of Problem 9, find a 95% confidence interval for the mean difference between percentage increase in company revenue and percentage increase in CEO salary. (b) Use the confidence interval method of hypothesis testing outlined in Problem 25 of Section 8.2 to test the hypothesis that population mean percentage increase in company revenue is different from that of CEO salary. Use a 5% level of significance. 9. Business: CEO Raises Are Americas top chief executive officers (CEOs) really worth all that money? One way to answer this question is to look at row B, the annual company percentage increase in revenue, versus row A, the CEOs annual percentage salary increase in that same company (Source: Forbes, Vol. 159, No. 10). A random sample of companies such as John Deere Co., General Electric, and Dow Chemical yielded the following data: Do these data indicate that the population mean percentage increase in corporate revenue (row B) is different from the population mean percentage increase in CEO salary? Use a 5% level of significance. 25. Expand Your Knowledge: Confidence Intervals and Two-Tailed Hypothesis Tests Is there a relationship between confidence intervals and two-tailed hypothesis tests? Let c be the level of confidence used to construct a confidence interval from sample data. Let a be the level of significance for a two-tailed hypothesis test. The following statement applies to hypothesis tests of the mean. For a two-tailed hypothesis test with level of significance and null hypothesis H0: = k, we reject H0 Whenever k falls outside the c = 1 confidence interval for based on the sample data When k falls within the c = 1 confidence interval, we do not reject H0. (A corresponding relationship between confidence intervals and two-tailed hypothesis tests also is valid for other parameters, such as p, 1 2, and p1 p2, which we will study in Sections 8.3 and 8.5.) Whenever the value of k given in the null hypothesis falls outside the c = 1 confidence interval for the parameter, we reject H0. For example, consider a two-tailed hypothesis test with = 0.01 and H0: = 20H1: 20 A random sample of size 36 has a sample mean x = 22 from a population with standard deviation = 4. (a) What is the value of c = 1 ? Using the methods of Chapter 7, construct a 1 confidence interval for from the sample data. What is the value of m given in the null hypothesis (i.e., what is k)? Is this value in the confidence interval? Do we reject or fail to reject H0 based on this information? (b) Using methods of this chapter, find the P-value for the hypothesis test. Do we reject or fail to reject H0? Compare your result to that of part (a).23P24PStatistical Literacy Consider a hypothesis test of difference of means for two independent populations x1 and x2. (a) What does the null hypothesis say about the relationship between the two population means? (b) If the sample test statistic has a z distribution, give the formula for z. (c) If the sample test statistic has a t distribution, give the formula for t.2P3PStatistical Literacy Consider a hypothesis test of difference of proportions for two independent populations. Suppose random samples produce r1 successes out of n1 trials for the first population and r2 successes out of n2 trials for the second population. (a) What does the null hypothesis claim about the relationship between the proportions of successes in the two populations? (b) What is the formula for the z value of the sample test statistic?Statistical Literacy Consider a hypothesis test of difference of proportions for two independent populations. Suppose random samples produce r1 successes out of n1 trials for the first population and r2 successes out of n2 trials for the second population. What is the best pooled estimate p for the population probability of success using H0: p1 = p2?Critical Thinking Consider use of a Students t distribution to test the difference of means for independent populations using random samples of sizes n1 and n2. (a) Which process gives the larger degrees of freedom, Satterthwaites approximation or using the smaller of n1 1 and n2 + 1? Which method is more conservative? What do we mean by conservative? Note that computer programs and other technologies commonly use Satterthwaites approximation. (b) Using the same hypotheses and sample data, is the P-value smaller for larger degrees of freedom? How might a larger P-value impact the significance of a test?Critical Thinking When conducting a test for the difference of means for two independent populations x1 and x2, what alternate hypothesis would indicate that the mean of the x2 population is smaller than that of the x1 population? Express the alternate hypothesis in two ways.Critical Thinking When conducting a test for the difference of means for two independent populations x1 and x2, what alternate hypothesis would indicate that the mean of the x2 population is larger than that of the x1 population? Express the alternate hypothesis in two ways.Basic Computation: Testing 1 2 A random sample of 49 measurements from one population had a sample mean of 10, with sample standard deviation 3. An independent random sample of 64 measurements from a second population had a sample mean of 12, with sample standard deviation 4. Test the claim that the population means are different. Use level of significance 0.01. (a) Check Requirements What distribution does the sample test statistic follow? Explain. (b) State the hypotheses. (c) Compute x1 x2 and the corresponding sample distribution value. (d) Estimate the P-value of the sample test statistic. (e) Conclude the test. (f) Interpret the results.Basic Computation: Testing 1 2 Two populations have mound-shaped, symmetric distributions. A random sample of 16 measurements from the first population had a sample mean of 20, with sample standard deviation 2. An independent random sample of 9 measurements from the second population had a sample mean of 19, with sample standard deviation 3. Test the claim that the population mean of the first population exceeds that of the second. Use a 5% level of significance. (a) Check Requirements What distribution does the sample test statistic follow? Explain. (b) State the hypotheses. (c) Compute x1 x2 and the corresponding sample distribution value. (d) Estimate the P-value of the sample test statistic. (e) Conclude the test. (f) Interpret the results.Basic Computation: Testing 1 2 A random sample of 49 measurements from a population with population standard deviation 3 had a sample mean of 10. An independent random sample of 64 measurements from a second population with population standard deviation 4 had a sample mean of 12. Test the claim that the population means are different. Use level of significance 0.01. (a) Check Requirements What distribution does the sample test statistic follow? Explain. (b) State the hypotheses. (c) Compute x1 x2 and the corresponding sample distribution value. (d) Find the P-value of the sample test statistic. (e) Conclude the test. (f) Interpret the results.Basic Computation: Testing 1 2 Two populations have normal distributions. The first has population standard deviation 2 and the second has population standard deviation 3. A random sample of 16 measurements from the first population had a sample mean of 20. An independent random sample of 9 measurements from the second population had a sample mean of 19. Test the claim that the population mean of the first population exceeds that of the second. Use a 5% level of significance. (a) Check Requirements What distribution does the sample test statistic follow? Explain. (b) State the hypotheses. (c) Compute x1 x2 and the corresponding sample distribution value. (d) Find the P-value of the sample test statistic. (e) Conclude the test (f) Interpret the results.13P14PPlease provide the following information for Problems 1526 and 2935. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding z or t value as appropriate. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value a small amount and therefore produce a slightly more conservative answer. Answers may vary due to rounding. 15. Medical: REM Sleep REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults (Reference: Secrets of Sleep by Dr. A. Borbely). Assume that REM sleep time is normally distributed for both children and adults. A random sample of n1 = 10 children (9 years old) showed that they had an average REM sleep time of x1=2.8 hours per night. From previous studies, it is known that 1 = 0.5 hour. Another random sample of n2 = 10 adults showed that they had an average REM sleep time of x2=2.1 hours per night. Previous studies show that 2 = 0.7 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance.Please provide the following information for Problems 1526 and 2935. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding z or t value as appropriate. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value a small amount and therefore produce a slightly more conservative answer. Answers may vary due to rounding. 16. Environment: Pollution Index Based on information from The Denver Post, a random sample of n1 = 12 winter days in Denver gave a sample mean pollution index of x1=43. Previous studies show that 1 = 21. For Englewood (a suburb of Denver), a random sample of n2= 14 winter days gave a sample mean pollution index of x2=36. Previous studies show that 2 = 15. Assume the pollution index is normally distributed in both Englewood and Denver. Do these data indicate that the mean population pollution index of Englewood is different (either way) from that of Denver in the winter? Use a 1% level of significance.Please provide the following information for Problems 1526 and 2935. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding z or t value as appropriate. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value a small amount and therefore produce a slightly more conservative answer. Answers may vary due to rounding. 17. Survey: Outdoor Activities A Michigan study concerning preference for outdoor activities used a questionnaire with a 6-point Likert-type response in which 1 designated not important and 6 designated extremely important. A random sample of n1 = 46 adults were asked about fishing as an outdoor activity. The mean response was x1=4.9. Another random sample of n2 = 51 adults were asked about camping as an outdoor activity. For this group, the mean response was x2=4.3. From previous studies, it is known that 1 = 1.5 and 2 = 1.2. Does this indicate a difference (either way) regarding preference for camping versus preference for fishing as an outdoor activity? Use a 5% level of significance. Note: A Likert scale usually has to do with approval of or agreement with a statement in a questionnaire. For example, respondents are asked to indicate whether they strongly agree, agree, disagree, or strongly disagree with the statement.Please provide the following information for Problems 1526 and 2935. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding z or t value as appropriate. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value a small amount and therefore produce a slightly more conservative answer. Answers may vary due to rounding. 18. Generation Gap: Education Education influences attitude and lifestyle. Differences in education are a big factor in the generation gap. Is the younger generation really better educated? Large surveys of people age 65 and older were taken in n1 = 32 U.S. cities. The sample mean for these cities showed that x1=15.2 of the older adults had attended college. Large surveys of young adults (ages 2534) were taken in n2 = 35 U.S. cities. The sample mean for these cities showed that x2=19.7 of the young adults had attended college. From previous studies, it is known that 1 = 7.2% and 2 = 5.2% (Reference: American Generations by S. Mitchell). Does this information indicate that the population mean percentage of young adults who attended college is higher? Use = 0.05.Please provide the following information for Problems 1526 and 2935. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding z or t value as appropriate. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value a small amount and therefore produce a slightly more conservative answer. Answers may vary due to rounding. 19. Crime Rate: FBI A random sample of n1 = 10 regions in New England gave the following violent crime rates (per million population). x1: New England Crime Rate 3.53.74.03.93.34.11.84.82.93.1 Another random sample of n2 = 12 regions in the Rocky Mountain states gave the following violent crime rates (per million population). x2: Rocky Mountain States 3.74.34.55.33.34.83.52.43.13.55.22.8 (Reference: Crime in the United States, Federal Bureau of Investigation.) Assume that the crime rate distribution is approximately normal in both regions. i. Use a calculator to verify that x13.51,s10.81,x23.87, and s20.94. ii. Do the data indicate that the violent crime rate in the Rocky Mountain region is higher than that in New England? Use = 0.01.Please provide the following information for Problems 1526 and 2935. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding z or t value as appropriate. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value a small amount and therefore produce a slightly more conservative answer. Answers may vary due to rounding. 20. Medical: Hay Fever A random sample of n1 = 16 communities in western Kansas gave the following information for people under 25 years of age. x1: Rate of hay fever per 1000 population for people under 25 A random sample of n2 = 14 regions in western Kansas gave the following information for people over 50 years old. x2: Rate of hay fever per 1000 population for people over 50 (Reference: National Center for Health Statistics.) i. Use a calculator to verify that x1109.50,s115.41,x299.36, and s211.57. ii. Assume that the hay fever rate in each age group has an approximately normal distribution. Do the data indicate that the age group over 50 has a lower rate of hay fever? Use = 0.05.21PPlease provide the following information for Problems 1526 and 2935. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding z or t value as appropriate. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value a small amount and therefore produce a slightly more conservative answer. Answers may vary due to rounding. 22. Education: Tutoring In the article cited in Problem 21, the results of the following experiment were reported. Form 2 of the GatesMacGintie Reading Test was administered to both an experimental group and a control group after 6 weeks of instruction, during which the experimental group received peer tutoring and the control group did not. For the experimental group n1 = 30 children, the mean score on the vocabulary portion of the test was x1=368.4, with sample standard deviation s1 = 39.5. The average score on the vocabulary portion of the test for the n2 = 30 subjects in the control group was x2=349.2, with sample standard deviation s2 = 56.6. Use a 1% level of significance to test the claim that the experimental group performed better than the control group. 21. Education: Tutoring In the journal Mental Retardation, an article reported the results of a peer tutoring program to help mildly mentally retarded children learn to read. In the experiment, the mildly retarded children were randomly divided into two groups: the experimental group received peer tutoring along with regular instruction, and the control group received regular instruction with no peer tutoring. There were n1 = n2 = 30 children in each group. The GatesMacGintie Reading Test was given to both groups before instruction began. For the experimental group, the mean score on the vocabulary portion of the test was x1=344.5, with sample standard deviation s1 = 49.1. For the control group, the mean score on the same test was x2=354.2, with sample standard deviation s2 = 50.9. Use a 5% level of significance to test the hypothesis that there was no difference in the vocabulary scores of the two groups before the instruction began.23P24P25P26P27P28P29P30PPlease provide the following information for Problems 1526 and 2935. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding z or t value as appropriate. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value a small amount and therefore produce a slightly more conservative answer. Answers may vary due to rounding. 31. Sociology: High School Dropouts This problem is based on information taken from Life in Americas Fifty States by G. S. Thomas. A random sample of n1 = 153 people ages 16 to 19 was taken from the island of Oahu, Hawaii, and 12 were found to be high school dropouts. Another random sample of n2 = 128 people ages 16 to 19 was taken from Sweetwater County, Wyoming, and 7 were found to be high school dropouts. Do these data indicate that the population proportion of high school dropouts on Oahu is different (either way) from that of Sweetwater County? Use a 1% level of significance.32P33P34P35P36P37P38P1CRP2CRPCritical Thinking All other conditions being equal, does a larger sample size increase or decrease the corresponding magnitude of the z or t value of the sample test statistic?4CRPBefore you solve each problem below, first categorize it by answering the following question: Are we testing a single mean, a difference of means, a paired difference, a single proportion, or a difference of proportions? Assume underlying population distributions are mound-shaped and symmetric for problems with small samples that involve testing a mean or difference of means. Then provide the following information for Problems 518. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding distribution value. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more conservative answer. Answers may vary due to rounding. 5. Vehicles: Mileage Based on information in Statistical Abstract of the United States (116th edition), the average annual miles driven per vehicle in the United States is 11.1 thousand miles, with 600 miles. Suppose that a random sample of 36 vehicles owned by residents of Chicago showed that the average mileage driven last year was 10.8 thousand miles. Does this indicate that the average miles driven per vehicle in Chicago is different from (higher or lower than) the national average? Use a 0.05 level of significance.6CRP7CRP8CRP9CRP10CRP11CRP12CRP13CRP14CRPBefore you solve each problem below, first categorize it by answering the following question: Are we testing a single mean, a difference of means, a paired difference, a single proportion, or a difference of proportions? Assume underlying population distributions are mound-shaped and symmetric for problems with small samples that involve testing a mean or difference of means. Then provide the following information for Problems 518. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding distribution value. (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more conservative answer. Answers may vary due to rounding. 15. Vending Machines: Coffee A machine in the student lounge dispenses coffee. The average cup of coffee is supposed to contain 7.0 ounces. Eight cups of coffee from this machine show the average content to be 7.3 ounces with a standard deviation of 0.5 ounce. Do you think that the machine has slipped out of adjustment and that the average amount of coffee per cup is different from 7 ounces? Use a 5% level of significance.16CRP17CRP18CRP1DH2DHDiscuss each of the following topics in class or review the topics on your own. Then write a brief but complete essay in which you summarize the main points. Please include formulas and graphs as appropriate. The most important questions in life usually cannot be answered with absolute certainty. Many important questions are answered by giving an estimate and a measure of confidence in the estimate. This was the focus of Chapter 7. However, sometimes important questions must be answered in a more straightforward manner by a simple yes or no. Hypothesis testing is the statistical process of answering questions with a straightforward yes or no and providing an estimate of the risk in accepting the answer. Review and discuss type I and type II errors associated with hypothesis testing.2LC3LC4LC5LC1UTStatistical Literacy When drawing a scatter diagram, along which axis is the explanatory variable placed? Along which axis is the response variable placed?2P3P4P5P6P7P8P9PCritical Thinking: Lurking Variables Over the past 30 years in the United States, there has been a strong negative correlation between the number of infant deaths at birth and the number of people over age 65. (a) Is the fact that people are living longer causing a decrease in infant mortalities at birth? (b) What lurking variables might be causing the increase in one or both of the variables? Explain.11P12P13PHealth Insurance: Administrative Cost The following data are based on information from Domestic Affairs. Let x be the average number of employees in a group health insurance plan, and let y be the average administrative cost as a percentage of claims. (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or strong? positive or negative? (c) Use a calculator to verify that x = 135, x2 = 7133, y = 148, y2 = 4674, and xy = 3040. Compute r. As x increases from 3 to 75, does the value of r imply that y should tend to increase or decrease? Explain.15PGeology: Earthquakes Is the magnitude of an earthquake related to the depth below the surface at which the quake occurs? Let x be the magnitude of an earthquake (on the Richter scale), and let y be the depth (in kilometers) of the quake below the surface at the epicenter. The following is based on information taken from the National Earthquake Information Service of the U.S. Geological Survey. Additional data may be found by visiting the web site for the service. (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or strong? positive or negative? (c) Use a calculator to verify that x = 24.1, x2 = 85.75, y = 53.5, y2 = 458.31, and xy = 190.18. Compute r. As x increases, does the value of r imply that y should tend to increase or decrease? Explain.17P18P19P20P21P22P23P24PStatistical Literacy In the least-squares line y=52x, what is the value of the slope? When x changes by 1 unit, by how much does y change?2PCritical Thinking When we use a least-squares line to predict y values for x values beyond the range of x values found in the data, are we extrapolating or interpolating? Are there any concerns about such predictions?4P5PCritical Thinking: Interpreting Computer Printouts Refer to the description of a computer display for regression described in Problem 5. The following Minitab display gives information regarding the relationship between the body weight of a child (in kilograms) and the metabolic rate of the child (in 100 kcal/24 hr). The data is based on information from The Merck Manual (a commonly used reference in medical schools and nursing programs). (a) Write out the least-squares equation. (b) For each 1-kilogram increase in weight, how much does the metabolic rate of a child increase? (c) What is the value of the sample correlation coefficient r? (d) Interpretation What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained?7PFor Problems 718, please do the following. (a) Draw a scatter diagram displaying the data. (b) Verify the given sums x, y, x2, y2, and xy and the value of the sample correlation coefficient r. (c) Find x, y, a, and b. Then find the equation of the least-squares line y=a+bx. (d) Graph the least-squares line on your scatter diagram. Be sure to use the point (x,y) as one of the points on the line. (e) Interpretation Find the value of the coefficient of determination r2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? 8. Ranching: Cattle You are the foreman of the Bar-S cattle ranch in Colorado. A neighboring ranch has calves for sale, and you are going to buy some to add to the Bar-S herd. How much should a healthy calf weigh? Let x be the age of the calf (in weeks), and let y be the weight of the calf (in kilograms). The following information is based on data taken from The Merck Veterinary Manual (a reference used by many ranchers). Complete parts (a) through (e), given x = 92, y = 617, x2 = 2338, y2 = 82,389, xy = 13,642, and r 0.998. (f) The calves you want to buy are 12 weeks old. What does the least-squares line predict for a healthy weight?9PFor Problems 718, please do the following. (a) Draw a scatter diagram displaying the data. (b) Verify the given sums x, y, x2, y2, and xy and the value of the sample correlation coefficient r. (c) Find x, y, a, and b. Then find the equation of the least-squares line y=a+bx. (d) Graph the least-squares line on your scatter diagram. Be sure to use the point (x,y) as one of the points on the line. (e) Interpretation Find the value of the coefficient of determination r2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? 10. Basketball: Fouls Data for this problem are based on information from STATS Basketball Scoreboard. It is thought that basketball teams that make too many fouls in a game tend to lose the game even if they otherwise play well. Let x be the number of fouls that were more than (i.e., over and above) the number of fouls made the opposing team made. Let y be the percentage of times the team with the larger number of fouls won the game. Complete parts (a) through (e), given x = 13, y = 154, x2 = 65, y2 = 6290, xy = 411, and r 0.988. (f) If a team had x = 4 fouls over and above the opposing team, what does the least-squares equation forecast for y?11P12PFor Problems 718, please do the following. (a) Draw a scatter diagram displaying the data. (b) Verify the given sums x, y, x2, y2, and xy and the value of the sample correlation coefficient r. (c) Find x, y, a, and b. Then find the equation of the least-squares line y=a+bx. (d) Graph the least-squares line on your scatter diagram. Be sure to use the point (x,y) as one of the points on the line. (e) Interpretation Find the value of the coefficient of determination r2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? 13. Income: Medical Care Let x be per capita income in thousands of dollars. Let y be the number of medical doctors per 10,000 residents. Six small cities in Oregon gave the following information about x and y (based on information from Life in Americas Small Cities by G. S. Thomas, Prometheus Books). Complete parts (a) through (e), given x = 53, y = 83.7, x2 = 471.04, y2 = 1276.83, xy = 755.89, and r 0.934. (f) Suppose a small city in Oregon has a per capita income of 10 thousand dollars. What is the predicted number of M.D.s per 10,000 residents?14P15PFor Problems 718, please do the following. (a) Draw a scatter diagram displaying the data. (b) Verify the given sums x, y, x2, y2, and xy and the value of the sample correlation coefficient r. (c) Find x, y, a, and b. Then find the equation of the least-squares line y=a+bx. (d) Graph the least-squares line on your scatter diagram. Be sure to use the point (x,y) as one of the points on the line. (e) Interpretation Find the value of the coefficient of determination r2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? 16. Research: Patents The following data are based on information from the Harvard Business Review (Vol. 72, No. 1). Let x be the number of different research programs, and let y be the mean number of patents per program. As in any business, a company can spread itself too thin. For example, too many research programs might lead to a decline in overall research productivity. The following data are for a collection of pharmaceutical companies and their research programs: Complete parts (a) through (e), given x = 90, y = 8.1, x2 = 1420, y2 = 11.83, xy = 113.8, and r 0.973. (f) Suppose a pharmaceutical company has 15 different research programs. What does the least-squares equation forecast for y = mean number of patents per program?17P18P19PResidual Plot: Miles per Gallon Consider the data of Problem 9. (a) Make a residual plot for the least-squares model. (b) Use the residual plot to comment about the appropriateness of the least-squares model for these data. See Problem 19. 9. Weight of Car: Miles per Gallon Do heavier cars really use more gasoline? Suppose a car is chosen at random. Let x be the weight of the car (in hundreds of pounds), and let y be the miles per gallon (mpg). The following information is based on data taken from Consumer Reports (Vol. 62, No. 4). Complete parts (a) through (e), given x = 299, y = 167, x2 = 11,887, y2 = 3773, xy = 5814, and r 0.946. (f) Suppose a car weighs x = 38 (hundred pounds). What does the least-squares line forecast for y = miles per gallon? Expand Your Knowledge: Residual Plot The least-squares line usually does not go through all the sample data points (x, y). In fact, for a specified x value from a data pair (x, y), there is usually a difference between the predicted value and the y value paired with x. This difference is called the residual. The residual is the difference between the y value in a specified data pair (x, y) and the value y=a+bx predicted by the least-squares line for the same x. yy is the residual. One way to assess how well a least-squares line serves as a model for the data is a residual plot. To make a residual plot, we put the x values in order on the horizontal axis and plot the corresponding residuals yy in the vertical direction. Because the mean of the residuals is always zero for a least-squares model, we place a horizontal line at zero. The accompanying figure shows a residual plot for the data of Guided Exercise 4, in which the relationship between the number of ads run per week and the number of cars sold that week was explored. To make the residual plot, first compute all the residuals. Remember that x and y are the given data values, and y is computed from the least-squares line y6.56+1.01x. (a) If the least-squares line provides a reasonable model for the data, the pattern of points in the plot will seem random and unstructured about the horizontal line at 0. Is this the case for the residual plot? (b) If a point on the residual plot seems far outside the pattern of other points, it might reflect an unusual data point (x, y), called an outlier. Such points may have quite an influence on the least-squares model. Do there appear to be any outliers in the data for the residual plot?21P22P23P24P25P1P2P3P4P5P6P7PIn Problems 712, parts (a) and (b) relate to testing . Part (c) requests the value of Se. Parts (d) and (e) relate to confidence intervals for prediction. Parts (f) and (g) relate to testing and finding confidence intervals for . Answers may vary due to rounding. 8. Baseball: Batting Average and Strikeouts Let x be a random variable that represents the batting average of a professional baseball player. Let y be a random variable that represents the percentage of strikeouts of a professional baseball player. A random sample of n = 6 professional baseball players gave the following information (Reference: The Baseball Encyclopedia, Macmillan). (a) Verify that x = 1.842, y = 37.6, x2 = 0.575838, y2 = 290.78, xy = 10.87, and r0.891. (b) Use a 5% level of significance to test the claim that 0. (c) Verify that Se1.6838, a26.247, and b65.081. (d) Find the predicted percentage of strikeouts for a player with an x = 0.300 batting average. (e) Find an 80% confidence interval for y when x = 0.300. (f) Use a 5% level of significance to test the claim that 0. (g) Find a 90% confidence interval for and interpret its meaning.9P10PIn Problems 712, parts (a) and (b) relate to testing . Part (c) requests the value of Se. Parts (d) and (e) relate to confidence intervals for prediction. Parts (f) and (g) relate to testing and finding confidence intervals for . Answers may vary due to rounding. 11. Oceanography: Drift Rates Ocean currents are important in studies of climate change, as well as ecology studies of dispersal of plankton. Drift bottles are used to study ocean currents in the Pacific near Hawaii, the Solomon Islands, New Guinea, and other islands. Let x represent the number of days to recovery of a drift bottle after release and y represent the distance from point of release to point of recovery in km/100. The following data are taken from the reference by Professor E.A. Kay, University of Hawaii. Reference: A Natural History of the Hawaiian Islands, edited by E. A. Kay, University of Hawaii Press. (a) Verify that x = 492, y = 86.7, x2 = 65,546, y2 = 2030.55, xy = 11351.9, and r 0.93853. (b) Use a 1% level of significance to test the claim 0. (c) Verify that Se 4.5759, a 1.1405, and b 0.1646 (d) Find the predicted distance (km/100) when a drift bottle has been floating for 90 days. (e) Find a 90% confidence interval for your prediction of part (d). (f) Use a 1% level of significance to test the claim that 0. (g) Find a 95% confidence interval for and interpret its meaning in terms of drift rate. (h) Consider the following scenario. A sailboat had an accident and radioed a Mayday alert with a given latitude and longitude just before it sank. The survivors are in a small (but well provisioned) life raft drifting in the part of the Pacific Ocean under study. After 30 days, how far from the accident site should a rescue plane expect to look?12P13P14P15PExpand Your Knowledge: Time Series and Serial Correlation An Internet advertising agency is studying the number of hits on a certain web site during an advertising campaign. It is hoped that as the campaign progresses, the number of hits on the web site will also increase in a predictable way from one day to the next. For 10 days of the campaign, the number of hits 105 is shown: Original Time Series (a) To construct a serial correlation, we use data pairs (x, y) where x = original data and y = original data shifted ahead by one time period. Verify that the data set (x, y) for serial correlation is shown here. (For discussion of serial correlation, see Problem 15.) (b) For the (x, y) data set of part (a), compute the equation of the sample least-squares line y=a+bx. If the number of hits was 9.3(105) one day, what do you predict for the number of hits the next day? (c) Compute the sample correlation coefficient r and the coefficient of determination r2. Test 0 at the 1 % level of significance. Would you say the time series of web site hits is relatively predictable from one day to the next? Explain. 15. Expand Your Knowledge: Time Series and Serial Correlation Serial correlation, also known as autocorrelation, describes the extent to which the result in one period of a time series is related to the result in the next period. A time series with high serial correlation is said to be very predictable from one period to the next. If the serial correlation is low (or near zero), the time series is considered to be much less predictable. For more information about serial correlation, see the book Ibbotson SBBI published by Morningstar. A research veterinarian at a major university has developed a new vaccine to protect horses from West Nile virus. An important question is: How predictable is the buildup of antibodies in the horses blood after the vaccination is given? A large random sample of horses from Wyoming were given the vaccination. The average antibody buildup factor (as determined from blood samples) was measured each week after the vaccination for 8 weeks. Results are shown in the following time series: Original Time Series To construct a serial correlation, we simply use data pairs (x, y) where x = original buildup factor data and y = original data shifted ahead by 1 week. This gives us the following data set. Since we are shifting 1 week ahead, we now have 7 data pairs (not 8). Data for Serial Correlation For convenience, we are given the following sums: x = 48.6 y = 58.5 x2 = 383.84 y2 = 529.37 xy = 448.7 (a) Use the sums provided (or a calculator with least-squares regression) to compute the equation of the sample least-squares line, y=a+bx. If the buildup factor was x = 5.8 one week, what would you predict the buildup factor to be the next week? (b) Compute the sample correlation coefficient r and the coefficient of determination r2. Test 0 at the 1% level of significance. Would you say the time series of antibody buildup factor is relatively predictable from one week to the next? Explain.17PStatistical Literacy Given the linear regression equation x1=1.6+3.5x27.9x3+2.0x4 (a) Which variable is the response variable? Which variables are the explanatory variables? (b) Which number is the constant term? List the coefficients with their corresponding explanatory variables. (c) If x2 = 2, x3 = 1, and x4 = 5, what is the predicted value for x1? (d) Explain how each coefficient can be thought of as a slope under certain conditions. Suppose x3 and x4 were held at fixed but arbitrary values and x2 was increased by 1 unit. What would be the corresponding change in x1? Suppose x2 increased by 2 units. What would be the expected change in x1? Suppose x2 decreased by 4 units. What would be the expected change in x1? (e) Suppose that n = 12 data points were used to construct the given regression equation and that the standard error for the coefficient of x2 is 0.419. Construct a 90% confidence interval for the coefficient of x2. (f) Using the information of part (e) and level of significance 5%, test the claim that the coefficient of x2 is different from zero. Explain how the conclusion of this test would affect the regression equation.2PFor Problems 3-6, use appropriate multiple regression software of your choice and enter the data. Note that the data are also available for download at the Companion Sites for this text. 3. Medical: Blood Pressure The systolic blood pressure of individuals is thought to be related to both age and weight. For a random sample of 11 men, the following data were obtained: (a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation (see Section 3.2) for each variable. Relative to its mean, which variable has the greatest spread of data values? Which variable has the smallest spread of data values relative to its mean? (b) For each pair of variables, generate the sample correlation coefficient r. Compute the corresponding coefficient of determination r2. Which variable (other than x1) has the greatest influence (by itself) on x1? Would you say that both variables x2 and x3 show a strong influence on x1? Explain your answer. What percent of the variation in x1 can be explained by the corresponding variation in x2? Answer the same question for x3. (c) Perform a regression analysis with x1 as the response variable. Use x2 and x3 as explanatory variables. Look at the coefficient of multiple determination. What percentage of the variation in x1 can be explained by the corresponding variations in x3 and x3 taken together? (d) Look at the coefficients of the regression equation. Write out the regression equation. Explain how each coefficient can be thought of as a slope. If age were held fixed, but a person put on 10 pounds, what would you expect for the corresponding change in systolic blood pressure? If a person kept the same weight but got 10 years older, what would you expect for the corresponding change in systolic blood pressure? (e) Test each coefficient to determine if it is zero or not zero. Use level of significance 5%. Why would the outcome of each test help us determine whether or not a given variable should be used in the regression model? (f) Find a 90% confidence interval for each coefficient. (g) Suppose Michael is 68 years old and weighs 192 pounds. Predict his systolic blood pressure, and find a 90% confidence range for your prediction (if your software produces prediction intervals).For Problems 3-6, use appropriate multiple regression software of your choice and enter the data. Note that the data are also available for download at the Companion Sites for this text. 4. Education: Exam Scores Professor Gill has taught general psychology for many years. During the semester, she gives three multiple-choice exams, each worth 100 points. At the end of the course, Dr. Gill gives a comprehensive final worth 200 points. Let x1, x2, and x3 represent a students scores on exams 1, 2, and 3, respectively. Let x4 represent the students score on the final exam. Last semester Dr. Gill had 25 students in her class. The student exam scores are shown on the next page. Since Professor Gill has not changed the course much from last semester to the present semester, the preceding data should be useful for constructing a regression model that describes this semester as well. (a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation (see Section 3.2) for each variable. Relative to its mean, would you say that each exam had about the same spread of scores? Most professors do not wish to give an exam that is extremely easy or extremely hard. Would you say that all of the exams were about the same level of difficulty? (Consider both means and spread of test scores.) (b) For each pair of variables, generate the sample correlation coefficient r. Compute the corresponding coefficient of determination r2 Of the three exams 1,2, and 3, which do you think had the most influence on the final exam 4? Although one exam had more influence on the final exam, did the other two exams still have a lot of influence on the final? Explain each answer. (c) Perform a regression analysis with x4 as the response variable. Use x1 x2, and x3 as explanatory variables. Look at the coefficient of multiple determination. What percentage of the variation in x4 can be explained by the corresponding variations in x1 x2, and x3 taken together? (d) Write out the regression equation. Explain how each coefficient can be thought of as a slope. If a student were to study "extra hard" for exam 3 and increase his or her score on that exam by 10 points, what corresponding change would you expect on the final exam? (Assume that exams I and 2 remain "fixed" in their scores.) (e) Test each coefficient in the regression equation to determine if it is zero or not zero. Use level of significance 5%. Why would the outcome of each hypothesis test help us decide whether or not a given variable should be used in the regression equation? (f) Find a 90% confidence interval for each coefficient. (g) This semester Susan has scores of 68, 72, and 75 on exams 1, 2, and 3, respectively. Make a prediction for Susans score on the final exam and find a 90% confidence interval for your prediction (if your software supports prediction intervals).5P6P7P1CRP2CRP3CRP4CRP5CRP6CRP7CRP8CRP9CRP10CRP1DH1LC2LC3LC4LC5LC6LC1UT2UT3UT4UT5UT6UT7UTIn Problems 16, please use the following steps (i) through (v) for all hypothesis tests. (i) What is the level of significance? State the null and alternate hypotheses. (ii) Check Requirements What sampling distribution will you use? What assumptions are you making? What is the value of the sample test statistic? (iii) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (iv) Based on your answers in parts (i) to (iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (v) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not in the Students t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value a small amount and thereby produce a slightly more conservative answer. 1. Testing and Estimating , Known Let x be a random variable that represents micrograms of lead per liter of water (g/L . An industrial plant discharges water into a creek. The Environmental Protection Agency (EPA) has studied the discharged water and found x to have a normal distribution, with = 0.7 g/L (Reference: EPA Wetlands Case Studies). (a) The industrial plant says that the population mean value of x is = 2.0 g/L. However, a random sample of n = 10 water samples showed that x = 2.56 g/L. Does this indicate that the lead concentration population mean is higher than the industrial plant claims? Use =1%. (b) Find a 95% confidence interval for using the sample data and the EPA value for . (c) How large a sample should be taken to be 95% confident that the sample mean x is within a margin of error E = 0.2 g/L of the population mean?2CURP3CURP4CURP5CURP6CURP7CURP8CURPLinear Regression: Blood Glucose Let x be a random variable that represents blood glucose level after a 12-hour fast. Let y be a random variable representing blood glucose level 1 hour after drinking sugar water (after the 12-hour fast). Units are in milligrams per 10 milliliters (mg/10 ml). A random sample of eight adults gave the following information (Reference: American Journal of Clinical Nutrition, Vol. 19, pp. 345351). x = 63.8; x2 = 521.56; y = 90.7; y2 = 1070.87; xy = 739.65 (a) Draw a scatter diagram for the data. (b) Find the equation of the least-squares line and graph it on the scatter diagram. (c) Find the sample correlation coefficient r and the sample coefficient of determination r2. Explain the meaning of r2 in the context of the application. (d) If x = 9.0, use the least-squares line to predict y. Find an 80% confidence interval for your prediction. (e) Use level of significance 1% and test the claim that the population correlation coefficient is not zero. Interpret the results. (f) Find an 85% confidence interval for the slope of the population-based least-squares line. Explain its meaning in the context of the application.Statistical Literacy In general, are chi-square distributions symmetric or skewed? If skewed, are they skewed right or left?Statistical Literacy For chi-square distributions, as the number of degrees of freedom increases, does any skewness increase or decrease? Do chi-square distributions become more symmetric (and normal) as the number of degrees of freedom becomes larger and larger?3P4P5P6P7PInterpretation: Test of Independence Consider Charlottes study of source of fraud/identity theft and gender (see Problem 6). She computed sample 2 = 10.2. (a) How many degrees of freedom are used? Recall that there were 5 sources of fraud/identity theft and, of course, 2 genders. Approximate the P-value and conclude the test at the 5% level of significance. Would it seem that gender and source of fraud/identity theft are independent? (b) From this study, can Charlotte identify which source of fraud/identity theft is dependent with respect to gender? Explain.9PFor Problems 919, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. Are all the expected frequencies greater than 5? What sampling distribution will you use? What are the degrees of freedom? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence? (e) Interpret your conclusion in the context of the application. Use the expected values E to the hundredths place. Psychology: MyersBriggs The following table shows the MyersBriggs personality preferences for a random sample of 519 people in the listed professions (Atlas of Type Tables by Macdaid, McCaulley, and Kainz). T refers to thinking and F refers to feeling. Use the chi-square test to determine if the listed occupations and personality preferences are independent at the 0.01 level of significance.11PFor Problems 919, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. Are all the expected frequencies greater than 5? What sampling distribution will you use? What are the degrees of freedom? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence? (e) Interpret your conclusion in the context of the application. Use the expected values E to the hundredths place. Archaeology: Pottery The following table shows ceremonial ranking and type of pottery sherd for a random sample of 434 sherds at a location in the Sand Canyon Archaeological Project, Colorado (The Architecture of Social Integration in Prehistoric Pueblos, edited by Lipe and Hegmon). Use a chi-square test to determine if ceremonial ranking and pottery type are independent at the 0.05 level of significance.13P14P15P16P17PFor Problems 919, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. Are all the expected frequencies greater than 5? What sampling distribution will you use? What are the degrees of freedom? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence? (e) Interpret your conclusion in the context of the application. Use the expected values E to the hundredths place. Political Affiliation: Spending Two random samples were drawn from members of the U.S. Congress. One sample was taken from members who are Democrats and the other from members who are Republicans. For each sample, the number of dollars spent on federal projects in each congresspersons home district was recorded. (i) Make a cluster bar graph showing the percentages of Congress members from each party who spent each designated amount in their respective home districts. (ii) Use a 1% level of significance to test whether congressional members of each political party spent designated amounts in the same proportions.19PStatistical Literacy For a chi-square goodness-of-fit test, how are the degrees of freedom computed?2PStatistical Literacy Explain why goodness-of-fit tests are always right-tailed tests.4PFor Problems 516, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. Are all the expected frequencies greater than 5? What sampling distribution will you use? What are the degrees of freedom? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis that the population fits the specified distribution of categories? (e) Interpret your conclusion in the context of the application. Census: Age The age distribution of the Canadian population and the age distribution of a random sample of 455 residents in the Indian community of Red Lake Village (Northwest Territories) are shown below (based on U.S. Bureau of the Census, International Data Base). Use a 5% level of significance to test the claim that the age distribution of the general Canadian population fits the age distribution of the residents of Red Lake Village.For Problems 516, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. Are all the expected frequencies greater than 5? What sampling distribution will you use? What are the degrees of freedom? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis that the population fits the specified distribution of categories? (e) Interpret your conclusion in the context of the application. Census: Type of Household The type of household for the U.S. population and for a random sample of 411 households from the community of Dove Creek, Montana, are shown (based on Statistical Abstract of the United States). Use a 5% level of significance to test the claim that the distribution of U.S. households fits the Dove Creek distribution.7PFor Problems 516, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. Are all the expected frequencies greater than 5? What sampling distribution will you use? What are the degrees of freedom? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis that the population fits the specified distribution of categories? (e) Interpret your conclusion in the context of the application. Ecology: Deer The types of browse favored by deer are shown in the following table (The Mule Deer of Mesa Verde National Park, edited by Mierau and Schmidt). Using binoculars, volunteers observed the feeding habits of a random sample of 320 deer. Use a 5% level of significance to test the claim that the natural distribution of browse fits the deer feeding pattern.For Problems 516, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. Are all the expected frequencies greater than 5? What sampling distribution will you use? What are the degrees of freedom? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis that the population fits the specified distribution of categories? (e) Interpret your conclusion in the context of the application. Meteorology: Normal Distribution The following problem is based on information from the National Oceanic and Atmospheric Administration (NOAA) Environmental Data Service. Let x be a random variable that represents the average daily temperature (in degrees Fahrenheit) in July in the town of Kit Carson, Colorado. The x distribution has a mean of approximately 75F and standard deviation of approximately 8F. A 20-year study (620 July days) gave the entries in the rightmost column of the following table. (i) Remember that = 75 and = 8. Examine Figure 6-5 in Chapter 6. Write a brief explanation for Columns I, II, and III in the context of this problem. (ii) Use a 1% level of significance to test the claim that the average daily July temperature follows a normal distribution with = 75 and = 8.For Problems 516, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. Are all the expected frequencies greater than 5? What sampling distribution will you use? What are the degrees of freedom? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis that the population fits the specified distribution of categories? (e) Interpret your conclusion in the context of the application. Meteorology: Normal Distribution Let x be a random variable that represents the average daily temperature (in degrees Fahrenheit) in January for the town of Hana, Maui. The x variable has a mean of approximately 68F and standard deviation s of approximately 4F (see reference in Problem 9). A 20-year study (620 January days) gave the entries in the rightmost column of the following table. (i) Remember that = 68 and = 4. Examine Figure 6-5 in Chapter 6. Write a brief explanation for Columns I, II, and III in the context of this problem. (ii) Use a 1% level of significance to test the claim that the average daily January temperature follows a normal distribution with = 68 and = 4.For Problems 516, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. Are all the expected frequencies greater than 5? What sampling distribution will you use? What are the degrees of freedom? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis that the population fits the specified distribution of categories? (e) Interpret your conclusion in the context of the application. Ecology: Fish The Fish and Game Department stocked Lake Lulu with fish in the following proportions: 30% catfish, 15% bass, 40% bluegill, and 15% pike. Five years later it sampled the lake to see if the distribution of fish had changed. It found that the 500 fish in the sample were distributed as follows. In the 5-year interval, did the distribution of fish change at the 0.05 level?For Problems 516, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. Are all the expected frequencies greater than 5? What sampling distribution will you use? What are the degrees of freedom? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis that the population fits the specified distribution of categories? (e) Interpret your conclusion in the context of the application. Library: Book Circulation The director of library services at Fairmont College did a survey of types of books (by subject) in the circulation library. Then she used library records to take a random sample of 888 books checked out last term and classified the books in the sample by subject. The results are shown next. Using a 5% level of significance, test the claim that the subject distribution of books in the library fits the distribution of books checked out by students.For Problems 516, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. Are all the expected frequencies greater than 5? What sampling distribution will you use? What are the degrees of freedom? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis that the population fits the specified distribution of categories? (e) Interpret your conclusion in the context of the application. Census: California The accuracy of a census report on a city in southern California was questioned by some government officials. A random sample of 1215 people living in the city was used to check the report, and the results are shown here: Using a 1% level of significance, test the claim that the census distribution and the sample distribution agree14P15P16P17P18PStatistical Literacy Does the x distribution need to be normal in order to use the chi-square distribution to test the variance? Is it acceptable to use the chisquare distribution to test the variance if the x distribution is simply mound-shaped and more or less symmetric?Critical Thinking The x distribution must be normal in order to use a chi-square distribution to test the variance. What are some methods you can use to assess whether the x distribution is normal? Hint: See Chapter 6 and goodness-of-fit tests.3PFor Problems 311, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. What are the degrees of freedom? What assumptions are you making about the original distribution? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence? (e) Interpret your conclusion in the context of the application. (f) Find the requested confidence interval for the population variance or population standard deviation. Interpret the results in the context of the application. In each of the following problems, assume a normal population distribution. Sociology: Marriage The following problem is based on information from an article by N. Keyfitz in the American Journal of Sociology (Vol. 53, pp. 470480). Let x 5 age in years of a rural Quebec woman at the time of her first marriage. In the year 1941, the population variance of x was approximately 2 = 5.1. Suppose a recent study of age at first marriage for a random sample of 41 women in rural Quebec gave a sample variance s2 = 3.3. Use a 5% level of significance to test the claim that the current variance is less than 5.1. Find a 90% confidence interval for the population variance.For Problems 311, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. What are the degrees of freedom? What assumptions are you making about the original distribution? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence? (e) Interpret your conclusion in the context of the application. (f) Find the requested confidence interval for the population variance or population standard deviation. Interpret the results in the context of the application. In each of the following problems, assume a normal population distribution. Mountain Climbing: Accidents The following problem is based on information taken from Accidents in North American Mountaineering (jointly published by The American Alpine Club and The Alpine Club of Canada). Let x represent the number of mountain climbers killed each year. The long-term variance of x is approximately 2 = 136.2. Suppose that for the past 8 years, the variance has been s2 = 115.1. Use a 1% level of significance to test the claim that the recent variance for number of mountain climber deaths is less than 136.2. Find a 90% confidence interval for the population variance.For Problems 311, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. What are the degrees of freedom? What assumptions are you making about the original distribution? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence? (e) Interpret your conclusion in the context of the application. (f) Find the requested confidence interval for the population variance or population standard deviation. Interpret the results in the context of the application. In each of the following problems, assume a normal population distribution. Professors: Salaries The following problem is based on information taken from Academe, Bulletin of the American Association of University Professors. Let x represent the average annual salary of college and university professors (in thousands of dollars) in the United States. For all colleges and universities in the United States, the population variance of x is approximately 2 5 47.1. However, a random sample of 15 colleges and universities in Kansas showed that x has a sample variance s2 = 83.2. Use a 5% level of significance to test the claim that the variance for colleges and universities in Kansas is greater than 47.1. Find a 95% confidence interval for the population variance.For Problems 311, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. What are the degrees of freedom? What assumptions are you making about the original distribution? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence? (e) Interpret your conclusion in the context of the application. (f) Find the requested confidence interval for the population variance or population standard deviation. Interpret the results in the context of the application. In each of the following problems, assume a normal population distribution. Medical: Clinical Test A new kind of typhoid shot is being developed by a medical research team. The old typhoid shot was known to protect the population for a mean time of 36 months, with a standard deviation of 3 months. To test the time variability of the new shot, a random sample of 23 people were given the new shot. Regular blood tests showed that the sample standard deviation of protection times was 1.9 months. Using a 0.05 level of significance, test the claim that the new typhoid shot has a smaller variance of protection times. Find a 90% confidence interval for the population standard deviation.For Problems 311, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. What are the degrees of freedom? What assumptions are you making about the original distribution? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence? (e) Interpret your conclusion in the context of the application. (f) Find the requested confidence interval for the population variance or population standard deviation. Interpret the results in the context of the application. In each of the following problems, assume a normal population distribution. Veterinary Science: Tranquilizer Jim Mead is a veterinarian who visits a Vermont farm to examine prize bulls. In order to examine a bull, Jim first gives the animal a tranquilizer shot. The effect of the shot is supposed to last an average of 65 minutes, and it usually does. However, Jim sometimes gets chased out of the pasture by a bull that recovers too soon, and other times he becomes worried about prize bulls that take too long to recover. By reading journals, Jim has found that the tranquilizer should have a mean duration time of 65 minutes, with a standard deviation of 15 minutes. A random sample of 10 of Jims bulls had a mean tranquilized duration time of close to 65 minutes but a standard deviation of 24 minutes. At the 1% level of significance, is Jim justified in the claim that the variance is larger than that stated in his journal? Find a 95% confidence interval for the population standard deviation.For Problems 311, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. What are the degrees of freedom? What assumptions are you making about the original distribution? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence? (e) Interpret your conclusion in the context of the application. (f) Find the requested confidence interval for the population variance or population standard deviation. Interpret the results in the context of the application. In each of the following problems, assume a normal population distribution. Engineering: Jet Engines The fan blades on commercial jet engines must be replaced when wear on these parts indicates too much variability to pass inspection. If a single fan blade broke during operation, it could severely endanger a flight. A large engine contains thousands of fan blades, and safety regulations require that variability measurements on the population of all blades not exceed 2 = 0.18 mm2. An engine inspector took a random sample of 61 fan blades from an engine. She measured each blade and found a sample variance of 0.27 mm2. Using a 0.01 level of significance, is the inspector justified in claiming that all the engine fan blades must be replaced? Find a 90% confidence interval for the population standard deviation.For Problems 311, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. What are the degrees of freedom? What assumptions are you making about the original distribution? (c) Find or estimate the P-value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence? (e) Interpret your conclusion in the context of the application. (f) Find the requested confidence interval for the population variance or population standard deviation. Interpret the results in the context of the application. In each of the following problems, assume a normal population distribution. Law: Bar Exam A factor in determining the usefulness of an examination as a measure of demonstrated ability is the amount of spread that occurs in the grades. If the spread or variation of examination scores is very small, it usually means that the examination was either too hard or too easy. However, if the variance of scores is moderately large, then there is a definite difference in scores between better, average, and poorer students. A group of attorneys in a Midwest state has been given the task of making up this years bar examination for the state. The examination has 500 total possible points, and from the history of past examinations, it is known that a standard deviation of around 60 points is desirable. Of course, too large or too small a standard deviation is not good. The attorneys want to test their examination to see how good it is. A preliminary version of the examination (with slight modifications to protect the integrity of the real examination) is given to a random sample of 24 newly graduated law students. Their scores give a sample standard deviation of 72 points. (i) Using a 0.01 level of significance, test the claim that the population standard deviation for the new examination is 60 against the claim that the population standard deviation is different from 60. (ii) Find a 99% confidence interval for the population variance. (iii) Find a 99% confidence interval for the population standard deviation.11P1PStatistical Literacy When using the F distribution to test two variances, is it essential that each of the two populations be normally distributed? Would it be all right if the populations had distributions that were mound-shaped and more or less symmetric?3P4P5P6P7P8P