8-2. Exercises

1. BASIC

Ex 1. Find the rejection region (for the standardized test statistic) for each hypothesis test.

  1. H0:μ=27 vs. Ha:μ<27,@ α=0.05.H_0:μ=27 \space vs. \space H_a:μ<27, @\space α=0.05.

  2. H0:μ=52 vs. Ha:μ≠52,@ α=0.05.H_0:μ=52 \space vs. \space H_a:μ \ne 52, @ \spaceα=0.05.

  3. H0:μ=−105 vs. Ha:μ>−105,@ α=0.10.H_0:μ=-105 \space vs. \space H_a:μ> -105, @ \space α=0.10.

  4. H0:μ=78 vs. Ha:μ≠78,@ α=0.10.H_0:μ=78 \space vs. \space H_a:μ \ne 78, @ \spaceα=0.10.

Ex 2. Find the rejection region (for the standardized test statistic) for each hypothesis test.

  1. H0:μ=17 vs. Ha:μ<17,@ α=0.01.H_0:μ=17 \space vs. \space H_a:μ < 17, @ \spaceα=0.01.

  2. H0:μ=880 vs. Ha:μ≠880,@ α=0.01.H_0:μ=880 \space vs. \space H_a:μ \ne 880, @ \spaceα=0.01.

  3. H0:μ=−12 vs. Ha:μ>−12@ α=0.05.H_0:μ=-12 \space vs. \space H_a:μ > -12 @ \spaceα=0.05.

  4. H0:μ=21.1 vs. Ha:μ≠21.1,@ α=0.05.H_0:μ=21.1 \space vs. \space H_a:μ \ne 21.1, @ \spaceα=0.05.

Ex 3. Find the rejection region (for the standardized test statistic) for each hypothesis test. Identify the test as left-tailed, right-tailed, or two-tailed.

  1. H0:μ=141 vs. Ha:μ<141,@ α=0.20.H_0:μ=141 \space vs. \space H_a:μ < 141, @ \spaceα=0.20.

  2. H0:μ=−54 vs. Ha:μ<−54,@ α=0.05.H_0:μ=-54 \space vs. \space H_a:μ < -54, @ \spaceα=0.05.

  3. H0:μ=98 vs. Ha:μ≠98,@ α=0.05.H_0:μ=98 \space vs. \space H_a:μ \ne 98, @ \spaceα=0.05.

  4. H0:μ=3.8 vs. Ha:μ>3.8,@ α=0.001.H_0:μ=3.8 \space vs. \space H_a:μ > 3.8, @ \spaceα=0.001.

Ex 4. Find the rejection region (for the standardized test statistic) for each hypothesis test. Identify the test as left-tailed, right-tailed, or two-tailed.

  1. H0:μ=−62 vs. Ha:μ≠−62,@ α=0.005.H_0:μ=-62 \space vs. \space H_a:μ \ne -62, @ \spaceα=0.005.

  2. H0:μ=73 vs. Ha:μ>73,@ α=0.001.H_0:μ=73 \space vs. \space H_a:μ > 73, @ \spaceα=0.001.

  3. H0:μ=1124 vs. Ha:μ<1124,@ α=0.001.H_0:μ=1124 \space vs. \space H_a:μ < 1124, @ \spaceα=0.001.

  4. H0:μ=0.12 vs. Ha:μ≠0.12,@ α=0.001.H_0:μ=0.12 \space vs. \space H_a:μ \ne 0.12, @ \spaceα=0.001.

Ex 5. Compute the value of the test statistic for the indicated test, based on the information given.

  1. Testing H0:μ=72.2 vs. Ha:μ>72.2,σunknown,n=55,xˉ=75.1,s=9.25H_0:μ=72.2 \space vs. \space H_a:μ>72.2, σ unknown, n = 55, \bar{x}=75.1, s = 9.25.

  2. Testing H0:μ=58 vs. Ha:μ>58,σ=1.22,n=40,xˉ=58.5,s=1.29H_0:μ=58 \space vs. \space H_a:μ>58, σ=1.22, n = 40, \bar{x}=58.5, s = 1.29.

  3. Testing H0:μ=−19.5 vs. Ha:μ<−19.5,σ unknown,n=30,xˉ=23.2,s=9.55H_0:μ=-19.5 \space vs. \space H_a:μ<-19.5, σ \space unknown, n = 30, \bar{x}=23.2, s = 9.55.

  4. Testing H0:μ=805 vs. Ha:μ≠805,σ=37.5,n=75,xˉ=818,s=36.2H_0:μ=805 \space vs. \space H_a:μ\ne805, σ=37.5, n = 75, \bar{x}=818, s = 36.2 .

Ex 6. Compute the value of the test statistic for the indicated test, based on the information given.

  1. Testing H0:μ=342 vs. Ha:μ<342,σ=11.2,n=40,xˉ=339,s=10.3H_0:μ=342 \space vs. \space H_a:μ<342, σ=11.2, n = 40, \bar{x}=339, s = 10.3 .

  2. Testing H0:μ=105 vs. Ha:μ>105,σ=5.3,n=80,xˉ=107,s=5.1H_0:μ=105 \space vs. \space H_a:μ>105, σ=5.3, n = 80, \bar{x}=107, s = 5.1

  3. Testing H0:μ=−13.5 vs. Ha:μ≠−13.5,σ unknown,n=32,xˉ=−13.8,s=1.5H_0:μ=-13.5 \space vs. \space H_a:μ \ne -13.5, σ \space unknown, n = 32, \bar{x}=-13.8, s = 1.5

  4. Testing H0:μ=28 vs. Ha:μ≠28,σ unknown,n=68,xˉ=27.8,s=1.3H_0:μ=28 \space vs. \space H_a:μ \ne 28, σ \space unknown, n = 68, \bar{x}=27.8, s = 1.3.

Ex 7. Perform the indicated test of hypotheses, based on the information given.

  1. Test H0:μ=212 vs. Ha:μ<212,σ unknown,n=36,xˉ=211.2,s=2.2H_0:μ=212 \space vs. \space H_a:μ < 212, σ \space unknown, n = 36, \bar{x}=211.2, s = 2.2 .

  2. Test H0:μ=−18 vs. Ha:μ>−18,@α=0.05,σ=3.3,n=44,xˉ=−17.2,s=3.1H_0:μ=-18 \space vs. \space H_a:μ > -18, @ α=0.05, σ = 3.3, n = 44, \bar{x} =-17.2, s = 3.1.

  3. Test H0:μ=24 vs. Ha:μ≠24,@α=0.02,σ unknown,n=50,xˉ=22.8,s=1.9H_0:μ=24 \space vs. \space H_a:μ \ne 24, @ α=0.02, σ \space unknown, n = 50, \bar{x} =22.8, s = 1.9

Ex 8. Perform the indicated test of hypotheses, based on the information given.

  1. Test H0:μ=105 vs. Ha:μ>105,@α=0.05,σ unknown,n=30,xˉ=108,s=7.2H_0:μ=105 \space vs. \space H_a:μ > 105, @ α=0.05, σ \space unknown, n = 30, \bar{x} =108, s = 7.2.

  2. Test H0:μ=21 vs. Ha:μ<21,@α=0.01,σ unknown,n=78,xˉ=20.5,s=3.9H_0:μ=21 \space vs. \space H_a:μ < 21, @ α=0.01, σ \space unknown, n = 78, \bar{x} =20.5, s = 3.9

  3. Test H0:μ=−375 vs. Ha:μ≠−375,@α=0.01,σ=18.5,n=31,xˉ=−388,s=18.0H_0:μ=-375 \space vs. \space H_a:μ \ne -375, @ α=0.01, σ =18.5, n = 31, \bar{x} =-388, s = 18.0.

2. APPLICATIONS

Ex 9. In the past the average length of an outgoing telephone call from a business office has been 143 seconds. A manager wishes to check whether that average has decreased after the introduction of policy changes. A sample of 100 telephone calls produced a mean of 133 seconds, with a standard deviation of 35 seconds. Perform the relevant test at the 1% level of significance.

Ex 10. The government of an impoverished country reports the mean age at death among those who have survived to adulthood as 66.2 years. A relief agency examines 30 randomly selected deaths and obtains a mean of 62.3 years with standard deviation 8.1 years. Test whether the agency’s data support the alternative hypothesis, at the 1% level of significance, that the population mean is less than 66.2.

Ex 11. The average household size in a certain region several years ago was 3.14 persons. A sociologist wishes to test, at the 5% level of significance, whether it is different now. Perform the test using the information collected by the sociologist: in a random sample of 75 households, the average size was 2.98 persons, with sample standard deviation 0.82 person.

Ex 12. The recommended daily calorie intake for teenage girls is 2,200 calories/day. A nutritionist at a state university believes the average daily caloric intake of girls in that state to be lower. Test that hypothesis, at the 5% level of significance, against the null hypothesis that the population average is 2,200 calories/day using the following sample data: n=36,xˉ= 2,150,s=203n = 36, \bar{x}= 2,150, s = 203 .

Ex 13. An automobile manufacturer recommends oil change intervals of 3,000 miles. To compare actual intervals to the recommendation, the company randomly samples records of 50 oil changes at service facilities and obtains sample mean 3,752 miles with sample standard deviation 638 miles. Determine whether the data provide sufficient evidence, at the 5% level of significance, that the population mean interval between oil changes exceeds 3,000 miles.

Ex 14. A medical laboratory claims that the mean turn-around time for performance of a battery of tests on blood samples is 1.88 business days. The manager of a large medical practice believes that the actual mean is larger. A random sample of 45 blood samples yielded mean 2.09 and sample standard deviation 0.13 day. Perform the relevant test at the 10% level of significance, using these data.

Ex 15. A grocery store chain has as one standard of service that the mean time customers wait in line to begin checking out not exceed 2 minutes. To verify the performance of a store the company measures the waiting time in 30 instances, obtaining mean time 2.17 minutes with standard deviation 0.46 minute. Use these data to test the null hypothesis that the mean waiting time is 2 minutes versus the alternative that it exceeds 2 minutes, at the 10% level of significance.

Ex 16. A magazine publisher tells potential advertisers that the mean household income of its regular readership is $61,500. An advertising agency wishes to test this claim against the alternative that the mean is smaller. A sample of 40 randomly selected regular readers yields mean income $59,800 with standard deviation $5,850. Perform the relevant test at the 1% level of significance.

Ex 17. Authors of a computer algebra system wish to compare the speed of a new computational algorithm to the currently implemented algorithm. They apply the new algorithm to 50 standard problems; it averages 8.16 seconds with standard deviation 0.17 second. The current algorithm averages 8.21 seconds on such problems. Test, at the 1% level of significance, the alternative hypothesis that the new algorithm has a lower average time than the current algorithm.

Ex 18. A random sample of the starting salaries of 35 randomly selected graduates with bachelor’s degrees last year gave sample mean and standard deviation $41,202 and $7,621, respectively. Test whether the data provide sufficient evidence, at the 5% level of significance, to conclude that the mean starting salary of all graduates last year is less than the mean of all graduates two years before, $43,589.

3. ADDITIONAL EXERCISES

Ex 19. The mean household income in a region served by a chain of clothing stores is $48,750. In a sample of 40 customers taken at various stores the mean income of the customers was $51,505 with standard deviation $6,852.

  1. Test at the 10% level of significance the null hypothesis that the mean household income of customers of the chain is $48,750 against that alternative that it is different from $48,750.

  2. The sample mean is greater than $48,750, suggesting that the actual mean of people who patronize this store is greater than $48,750. Perform this test, also at the 10% level of significance. (The computation of the test statistic done in part (a) still applies here.)

Ex 20. The labor charge for repairs at an automobile service center are based on a standard time specified for each type of repair. The time specified for replacement of universal joint in a drive shaft is one hour. The manager reviews a sample of 30 such repairs. The average of the actual repair times is 0.86 hour with standard deviation 0.32 hour.

  1. Test at the 1% level of significance the null hypothesis that the actual mean time for this repair differs from one hour.

  2. The sample mean is less than one hour, suggesting that the mean actual time for this repair is less than one hour. Perform this test, also at the 1% level of significance. (The computation of the test statistic done in part (a) still applies here.)

4. LARGE DATA SET EXERCISES

Note: All of the data sets associated with these questions are missing, but the questions themselves are included here for reference.

Ex 21. Large Data Set 1 records the SAT scores of 1,000 students. Regarding it as a random sample of all high school students, use it to test the hypothesis that the population mean exceeds 1,510, at the 1% level of significance. (The null hypothesis is that μ=1510μ = 1510 .)

Ex 22. Large Data Set 1 records the GPAs of 1,000 college students. Regarding it as a random sample of all college students, use it to test the hypothesis that the population mean is less than 2.50, at the 10% level of significance. (The null hypothesis is that μ=2.50μ = 2.50 .)

Ex 23. Large Data Set 1 lists the SAT scores of 1,000 students.

  1. Regard the data as arising from a census of all students at a high school, in which the SAT score of every student was measured. Compute the population mean μ.

  2. Regard the first 50 students in the data set as a random sample drawn from the population of part (a) and use it to test the hypothesis that the population mean exceeds 1,510, at the 10% level of significance. (The null hypothesis is that μ=1510μ = 1510 .)

  3. Is your conclusion in part (b) in agreement with the true state of nature (which by part (a) you know), or is your decision in error? If your decision is in error, is it a Type I error or a Type II error?

Ex 24. Large Data Set 1 lists the GPAs of 1,000 students.

  1. Regard the data as arising from a census of all freshman at a small college at the end of their first academic year of college study, in which the GPA of every such person was measured. Compute the population mean μ.

  2. Regard the first 50 students in the data set as a random sample drawn from the population of part (a) and use it to test the hypothesis that the population mean is less than 2.50, at the 10% level of significance. (The null hypothesis is that μ=2.50μ = 2.50 .)

  3. Is your conclusion in part (b) in agreement with the true state of nature (which by part (a) you know), or is your decision in error? If your decision is in error, is it a Type I error or a Type II error?

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