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31. A new weight-watching company, Weight Reducers International, advertises that those who join will lose, on the average, 10 pounds the first two weeks with a standard deviation of 2.8 pounds. A random sample of 50 people who joined the new weight reduction program revealed the mean loss to be 9 pounds. At the .05 level of significance, can we conclude that those joining Weight Reducers on average will lose less than 10 pounds? Determine the p-value. π»π: π β₯ 10 π»π: π < 10 9β10 Test statistic: π§ = 2.8/β50 = β2.525 Critical value: π§ππ = β1.645 Reject the null hypothesis. We have sufficient evidence to conclude that those joining Weight Reducers on average will lose less than 10 pounds. p-value = 0.0058 32. Dole Pineapple, Inc., is concerned that the 16-ounce can of sliced pineapple is being overfilled. Assume the standard deviation of the process is .03 ounces. The qualitycontrol department took a random sample of 50 cans and found that the arithmetic mean weight was 16.05 ounces. At the 5 percent level of significance, can we conclude that the mean weight is greater than 16 ounces? Determine the p-value. π»π: π β€ 16 π»π: π > 16 Test statistic: π§ = 16.05β16 0.03/β50 = 11.785 Critical value: π§ππ = 1.645 Reject the null hypothesis. We have sufficient evidence to conclude that the mean weight is greater than 16 ounces. p-value = 0.0000 38. A recent article in The Wall Street Journal reported that the 30-year mortgage rate is now less than 6 percent. A sample of eight small banks in the Midwest revealed the following 30-year rates (in percent): 4.8 5.3 6.5 4.8 6.1 5.8 6.2 5.6 At the .01 significance level, can we conclude that the 30-year mortgage rate for small banks is less than 6 percent? Estimate the p-value. π»π: π β₯ 6 π»π: π < 6 π₯Μ = 5.638 π = 0.635 5.638β6 Test statistic: π‘ = 0.635/β8 = β1.616 degree of freedom = 8 β 1 = 7 Critical value: π‘ππ = β2.998 We cannot reject the null hypothesis. We donβt have sufficient evidence to conclude that the 30-year mortgage rate for small banks is less than 6 percent. p-value = 0.0751 27. A recent study focused on the number of times men and women who live alone buy take-out dinner in a month. The information is summarized below. Statistic Sample mean Population standard deviation Sample size Men 24.51 4.48 35 Women 22.69 3.86 40 At the .01 significance level, is there a difference in the mean number of times men and women order take-out dinners in a month? What is the p-value? π»π: π1 β π2 = 0 π»π: π1 β π2 β 0 Test statistic: π§ = 24.51β22.69 2 2 β4.48 +3.86 35 = 1.871 40 Critical value: π§ππ = 2.326 We cannot reject the null hypothesis. We donβt have sufficient evidence to conclude that there is a difference in the mean number of times men and women order take-out dinners in a month. p-value = 0.031 46. Grand Strand Family Medical Center is specifically set up to treat minor medical emergencies for visitors to the Myrtle Beach area. There are two facilities, one in the Little River Area and the other in Murrells Inlet. The Quality Assurance Department wishes to compare the mean waiting time for patients at the two locations. Samples of the waiting times, reported in minutes, follow: Location Waiting Time Little River 31.73 28.77 29.53 22.08 29.47 18.60 32.94 25.18 29.82 26.49 Murrells Inlet 22.93 23.92 26.92 27.20 26.44 25.62 30.61 29.44 23.09 23.10 26.69 22.31 Assume the population standard deviations are not the same. At the .05 significance level, is there a difference in the mean waiting time? π»π: π1 β π2 = 0 π»π: π1 β π2 β 0 π₯Μ 1 = 27.461 π 1 = 4.440 π1 = 10 π₯Μ 2 = 25.689 π 2 = 2.685 π2 = 12 Test statistic: π‘ = 27.461β25.689 2 β4.440 +2.685 10 2 = 1.105 12 degree of freedom = 14 Critical value: π‘ππ = β2.145 We cannot reject the null hypothesis. We donβt have sufficient evidence to conclude that there is a difference in the mean waiting times. 52. The president of the American Insurance Institute wants to compare the yearly costs of auto insurance offered by two leading companies. He selects a sample of 15 families, some with only a single insured driver, others with several teenage drivers, and pays each family a stipend to contact the two companies and ask for a price quote. To make the data comparable, certain features, such as the deductible amount and limits of liability, are standardized. The sample information is reported below. At the .10 significance level, can we conclude that there is a difference in the amounts quoted? Family Becker Berry Cobb Progressive Car Insurance $2,090 1,683 1,402 GEICO Mutual Insurance $1,610 1,247 2,327 Debuck DuBrul Eckroate German Glasson King Kucic Meredith Obeid Price Phillips Tresize 1,830 930 697 1,741 1,129 1,018 1,881 1,571 874 1,579 1,577 860 1,367 1,461 1,789 1,621 1,914 1,956 1,772 1,375 1,527 1,767 1,636 1,188 π»π: ππ· = 0 π»π: ππ· β 0 π₯Μ π· = 246.33 π π· = 546.96 246.33β0 Test statistic: π‘ = 546.96/β15 = 1.744 Degree of freedom = 15 β 1 = 14 Critical value: π‘ππ = 1.761 We cannot reject the null hypothesis. We donβt have sufficient evidence to conclude that there is a difference in the amounts quoted. 23. A real estate agent in the coastal area of Georgia wants to compare the variation in the selling price of homes on the oceanfront with those one to three blocks from the ocean. A sample of 21 oceanfront homes sold within the last year revealed the standard deviation of the selling prices was $45,600. A sample of 18 homes, also sold within the last year, that were one to three blocks from the ocean revealed that the standard deviation was $21,330. At the .01 significance level, can we conclude that there is more variation in the selling prices of the oceanfront homes? π»π: π12 β€ π22 π»π: π12 > π22 45,6002 Test statistic: πΉ = 21,3302 = 4.57 degree of freedom numerator = 21 β 1 = 20 degree of freedom denominator 18 β 1 = 17 Critical value: πΉππ = 3.607 We reject the null hypothesis. We have sufficient evidence to conclude that there is more variation in the selling prices of the oceanfront homes. 28. The following is a partial ANOVA table. Sum of Mean Squares 320 280 500 Source Treatment Error Total df 2 9 11 Square 160 20 F 8 Complete the table and answer the following questions. Use the .05 significance level. a. How many treatments are there? b. What is the total sample size? c. What is the critical value of F? d. Write out the null and alternate hypotheses. e. What is your conclusion regarding the null hypothesis? a. 3 b. 12 c. 4.256 d. Ho: ΞΌ1 = ΞΌ2 = ΞΌ3 Ha: at least one of the treatment means is different e. Reject the null hypothesis. 19. In a particular market there are three commercial television stations, each with its own evening news program from 6:00 to 6:30 P.M. According to a report in this morningβs local newspaper, a random sample of 150 viewers last night revealed 53 watched the news on WNAE (channel 5), 64 watched on WRRN (channel 11), and 33 on WSPD (channel 13). At the .05 significance level, is there a difference in the proportion of viewers watching the three channels? Ho: There is no difference Ha: There is difference Expected number of viewers = Test statistic: π 2 = (53β50)2 50 + 150 3 = 50 (64β50)2 50 + (33β50)2 50 = 9.88 Degree of freedom = 3 β 1 = 2 2 Critical value: πππ = 5.991 We reject the null hypothesis. We have sufficient evidence to conclude that there is a difference in the proportion of viewers watching the three channels. 20. There are four entrances to the Government Center Building in downtown Philadelphia. The building maintenance supervisor would like to know if the entrances are equally utilized. To investigate, 400 people were observed entering the building. The number using each entrance is reported below. At the .01 significance level, is there a difference in the use of the four entrances? Entrance Main Street Broad Street Cherry Street Walnut Street Total Frequency 140 120 90 50 400 Ho: There is no difference Ha: There is difference Expected number of viewers = Test statistic: π 2 = (140β100)2 100 + 400 4 = 100 (120β100)2 100 + (90β100)2 100 + (50β100)2 100 = 46 Degree of freedom = 4 β 1 = 3 2 Critical value: πππ = 11.34 We reject the null hypothesis. We have sufficient evidence to conclude that there is a difference in the use of the four entrances.