9-1. Exercises

1. BASIC

Ex 1. Construct the confidence interval for (μ1−μ2)(μ_1−μ_2) for the level of confidence and the data from independent samples given.

  1. 90% confidence,

    n1=45,x1ˉ=27,s1=2,n_1=45, \bar{x_1}=27, s_1=2 , n2=60,x2ˉ=22,s2=3n_2=60, \bar{x_2}=22, s_2=3

  2. 99% confidence,

    n1=30,x1ˉ=−112,s1=9,n_1=30, \bar{x_1}=-112, s_1=9 , n2=40,x2ˉ=−98,s2=4n_2=40, \bar{x_2}=-98, s_2=4

Ex 2. Construct the confidence interval for (μ1−μ2)(μ_1−μ_2) for the level of confidence and the data from independent samples given.

  1. 95% confidence,

    n1=110,x1ˉ=77,s1=15,n_1=110, \bar{x_1}=77, s_1=15,

    n2=85,x2ˉ=79,s2=21n_2=85, \bar{x_2}=79, s_2=21

  2. 90% confidence,

    n1=65,x1ˉ=−83,s1=12,n_1=65, \bar{x_1}=-83, s_1=12 ,

    n2=65,x2ˉ=−74,s2=8n_2=65, \bar{x_2}=-74, s_2=8

Ex 3. Construct the confidence interval for (μ1−μ2)(μ_1−μ_2) for the level of confidence and the data from independent samples given.

  1. 99.5% confidence,

    n1=130,x1ˉ=27.2,s1=2.5,n_1=130, \bar{x_1}=27.2, s_1=2.5 ,

    n2=155,x2ˉ=38.8,s2=4.6n_2=155, \bar{x_2}=38.8, s_2=4.6

  2. 95% confidence,

    n1=68,x1ˉ=215.5,s1=12.3,n_1=68, \bar{x_1}=215.5, s_1=12.3 ,

    n2=84,x2ˉ=287.8,s2=14.1n_2=84, \bar{x_2}=287.8, s_2=14.1

Ex 4. Construct the confidence interval for (μ1−μ2)(μ_1−μ_2) for the level of confidence and the data from independent samples given.

  1. 99.9% confidence,

    n1=275,x1ˉ=70.2,s1=1.5,n_1=275, \bar{x_1}=70.2, s_1=1.5,

    n2=325,x2ˉ=63.4,s2=1.1n_2=325, \bar{x_2}=63.4, s_2=1.1

  2. 90% confidence,

    n1=120,x1ˉ=35.5,s1=0.75,n_1=120, \bar{x_1}=35.5, s_1=0.75,

    n2=146,x2ˉ=29.6,s2=0.80n_2=146, \bar{x_2}=29.6, s_2=0.80

Ex 5. Perform the test of hypotheses indicated, using the data from independent samples given. Use the critical value approach. Compute the p-value of the test as well.

  1. Test H0:μ1−μ2=3 vs. Ha:μ1−μ2≠3,@α=0.05,H_0:μ1−μ2=3 \space vs. \space H_a:μ1−μ2≠3, @ α=0.05,

    n1=35,x1ˉ=25,s1=1,n_1=35, \bar{x_1}=25, s_1=1 ,

    n2=45,x2ˉ=19,s2=2n_2=45, \bar{x_2}=19, s_2=2

  2. Test H0:μ1−μ2=−25 vs. Ha:μ1−μ2<−25,@α=0.10,H_0:μ1−μ2=-25 \space vs. \space H_a:μ1−μ2<-25, @ α=0.10,

    n1=85,x1ˉ=188,s1=15,n_1=85, \bar{x_1}=188, s_1=15 ,

    n2=62,x2ˉ=215,s2=19n_2=62, \bar{x_2}=215, s_2=19

Ex 6. Perform the test of hypotheses indicated, using the data from independent samples given. Use the critical value approach. Compute the p-value of the test as well.

  1. Test H0:μ1−μ2=45 vs. Ha:μ1−μ2>45,@α=0.001,H_0:μ1−μ2=45 \space vs. \space H_a:μ1−μ2>45, @ α=0.001,

    n1=200,x1ˉ=1312,s1=35,n_1=200, \bar{x_1}=1312, s_1=35 , n2=225,x2ˉ=1256,s2=28n_2=225, \bar{x_2}=1256, s_2=28

  2. Test H0:μ1−μ2=−12 vs. Ha:μ1−μ2≠−12,@α=0.10,H_0:μ1−μ2=-12 \space vs. \space H_a:μ1−μ2≠-12, @ α=0.10,

    n1=35,x1ˉ=121,s1=6,n_1=35, \bar{x_1}=121, s_1=6 ,

    n2=40,x2ˉ=135,s2=7n_2=40, \bar{x_2}=135, s_2=7

Ex 7. Perform the test of hypotheses indicated, using the data from independent samples given. Use the critical value approach. Compute the p-value of the test as well.

  1. Test H0:μ1−μ2=0 vs. Ha:μ1−μ2≠0,@α=0.01,H_0:μ1−μ2=0 \space vs. \space H_a:μ1−μ2≠0, @ α=0.01,

    n1=125,x1ˉ=−46,s1=10,n_1=125, \bar{x_1}=-46, s_1=10,

    n2=90,x2ˉ=−50,s2=13n_2=90, \bar{x_2}=-50, s_2=13

  2. Test H0:μ1−μ2=20 vs. Ha:μ1−μ2>20,@α=0.05,H_0:μ1−μ2=20 \space vs. \space H_a:μ1−μ_2>20, @ α=0.05,

    n1=40,x1ˉ=142,s1=11,n_1=40, \bar{x_1}=142, s_1=11,

    n2=40,x2ˉ=118,s2=10n_2=40, \bar{x_2}=118, s_2=10

Ex 8. Perform the test of hypotheses indicated, using the data from independent samples given. Use the critical value approach. Compute the p-value of the test as well.

  1. Test H0:μ1−μ2=13 vs. Ha:μ1−μ2<13,@α=0.01,H_0:μ1−μ2=13 \space vs. \space H_a:μ1−μ2<13, @ α=0.01,

    n1=35,x1ˉ=100,s1=2,n_1=35, \bar{x_1}=100, s_1=2 ,

    n2=35,x2ˉ=88,s2=2n_2=35, \bar{x_2}=88, s_2=2

  2. Test H0:μ1−μ2=10 vs. Ha:μ1−μ2≠10,@α=0.10,H_0:μ1−μ2=10 \space vs. \space H_a:μ1−μ2≠10, @ α=0.10,

    n1=146,x1ˉ=62,s1=4,n_1=146, \bar{x_1}=62, s_1=4,

    n2=120,x2ˉ=73,s2=7n_2=120, \bar{x_2}=73, s_2=7

Ex 9. Perform the test of hypotheses indicated, using the data from independent samples given. Use the p-value approach.

  1. Test H0:μ1−μ2=57 vs. Ha:μ1−μ2<57,@α=0.10,H_0:μ1−μ2=57 \space vs. \space H_a:μ1−μ2<57, @ α=0.10,

    n1=171,x1ˉ=1309,s1=42,n_1=171, \bar{x_1}=1309, s_1=42, n2=133,x2ˉ=1258,s2=37n_2=133, \bar{x_2}=1258, s_2=37

  2. Test H0:μ1−μ2=−1.5 vs. Ha:μ1−μ2≠−1.5,@α=0.20,H_0:μ1−μ2=-1.5 \space vs. \space H_a:μ1−μ2≠-1.5, @ α=0.20,

    n1=65,x1ˉ=16.9,s1=1.3,n_1=65, \bar{x_1}=16.9, s_1=1.3, n2=57,x2ˉ=18.6,s2=1.1n_2=57, \bar{x_2}=18.6, s_2=1.1

Ex 10. Perform the test of hypotheses indicated, using the data from independent samples given. Use the p-value approach.

  1. Test H0:μ1−μ2=−10.5 vs. Ha:μ1−μ2>−10.5,@α=0.01,H_0:μ1−μ2=-10.5 \space vs. \space H_a:μ1−μ2>-10.5, @ α=0.01,

    n1=64,x1ˉ=85.6,s1=2.4,n_1=64, \bar{x_1}=85.6, s_1=2.4,

    n2=50,x2ˉ=95.3,s2=3.1n_2=50, \bar{x_2}=95.3, s_2=3.1

  2. Test H0:μ1−μ2=110 vs. Ha:μ1−μ2≠110,@α=0.02,H_0:μ1−μ2=110 \space vs. \space H_a:μ1−μ2≠110, @ α=0.02,

    n1=176,x1ˉ=1918,s1=68,n_1=176, \bar{x_1}=1918, s_1=68,

    n2=241,x2ˉ=1782,s2=146n_2=241, \bar{x_2}=1782, s_2=146

Ex 11. Perform the test of hypotheses indicated, using the data from independent samples given. Use the p-value approach.

  1. Test H0:μ1−μ2=50 vs. Ha:μ1−μ2>50,@α=0.005,H_0:μ1−μ2=50 \space vs. \space H_a:μ1−μ2>50, @ α=0.005,

    n1=72,x1ˉ=272,s1=26,n_1=72, \bar{x_1}=272, s_1=26,

    n2=103,x2ˉ=213,s2=14n_2=103, \bar{x_2}=213, s_2=14

  2. Test H0:μ1−μ2=7.5 vs. Ha:μ1−μ2≠7.5,@α=0.10,H_0:μ1−μ2=7.5 \space vs. \space H_a:μ1−μ2≠7.5, @ α=0.10,

    n1=52,x1ˉ=94.3,s1=2.6,n_1=52, \bar{x_1}=94.3, s_1=2.6,

    n2=38,x2ˉ=88.6,s2=8.0n_2=38, \bar{x_2}=88.6, s_2=8.0

Ex 12. Perform the test of hypotheses indicated, using the data from independent samples given. Use the p-value approach.

  1. Test H0:μ1−μ2=23 vs. Ha:μ1−μ2<23,@α=0.20,H_0:μ1−μ2=23 \space vs. \space H_a:μ1−μ2<23, @ α=0.20,

    n1=314,x1ˉ=198,s1=12.2,n_1=314, \bar{x_1}=198, s_1=12.2,

    n2=220,x2ˉ=176,s2=11.5n_2=220, \bar{x_2}=176, s_2=11.5

  2. Test H0:μ1−μ2=4.4 vs. Ha:μ1−μ2≠4.4,@α=0.05,H_0:μ1−μ2=4.4 \space vs. \space H_a:μ1−μ2≠4.4, @ α=0.05,

    n1=32,x1ˉ=40.3,s1=0.5,n_1=32, \bar{x_1}=40.3, s_1=0.5,

    n2=30,x2ˉ=35.5,s2=0.7n_2=30, \bar{x_2}=35.5, s_2=0.7

2. APPLICATIONS

Ex 13. In order to investigate the relationship between mean job tenure in years among workers who have a bachelor’s degree or higher and those who do not, random samples of each type of worker were taken, with the following results.

n

xˉ\bar{x}

s

Bachelor’s degree or higher

155

5.2

1.3

No degree

210

5.0

1.5

  1. Construct the 99% confidence interval for the difference in the population means based on these data.

  2. Test, at the 1% level of significance, the claim that mean job tenure among those with higher education is greater than among those without, against the default that there is no difference in the means.

  3. Compute the observed significance of the test.

Ex 14. Records of 40 used passenger cars and 40 used pickup trucks (none used commercially) were randomly selected to investigate whether there was any difference in the mean time in years that they were kept by the original owner before being sold. For cars the mean was 5.3 years with standard deviation 2.2 years. For pickup trucks the mean was 7.1 years with standard deviation 3.0 years.

  1. Construct the 95% confidence interval for the difference in the means based on these data.

  2. Test the hypothesis that there is a difference in the means against the null hypothesis that there is no difference. Use the 1% level of significance.

  3. Compute the observed significance of the test in part (b).

Ex 15. In previous years the average number of patients per hour at a hospital emergency room on weekends exceeded the average on weekdays by 6.3 visits per hour. A hospital administrator believes that the current weekend mean exceeds the weekday mean by fewer than 6.3 hours.

  1. Construct the 99% confidence interval for the difference in the population means based on the following data, derived from a study in which 30 weekend and 30 weekday one-hour periods were randomly selected and the number of new patients in each recorded.

    n

    xˉ\bar{x}

    s

    Weekends

    30

    13.8

    3.1

    Weekdays

    30

    8.6

    2.7

  2. Test at the 5% level of significance whether the current weekend mean exceeds the weekday mean by fewer than 6.3 patients per hour.

  3. Compute the observed significance of the test.

Ex 16. A sociologist surveys 50 randomly selected citizens in each of two countries to compare the mean number of hours of volunteer work done by adults in each. Among the 50 inhabitants of Lilliput, the mean hours of volunteer work per year was 52, with standard deviation 11.8. Among the 50 inhabitants of Blefuscu, the mean number of hours of volunteer work per year was 37, with standard deviation 7.2.

  1. Construct the 99% confidence interval for the difference in mean number of hours volunteered by all residents of Lilliput and the mean number of hours volunteered by all residents of Blefuscu.

  2. Test, at the 1% level of significance, the claim that the mean number of hours volunteered by all residents of Lilliput is more than ten hours greater than the mean number of hours volunteered by all residents of Blefuscu.

  3. Compute the observed significance of the test in part (b).

Ex 17. A university administrator asserted that upperclassmen spend more time studying than underclassmen.

  1. Test this claim against the default that the average number of hours of study per week by the two groups is the same, using the following information based on random samples from each group of students. Test at the 1% level of significance.

    n

    xˉ\bar{x}

    s

    Upperclassmen

    35

    15.6

    2.9

    Underclassmen

    35

    12.3

    4.1

  2. Compute the observed significance of the test.

Ex 18. An kinesiologist claims that the resting heart rate of men aged 18 to 25 who exercise regularly is more than five beats per minute less than that of men who do not exercise regularly. Men in each category were selected at random and their resting heart rates were measured, with the results shown.

n

xˉ\bar{x}

s

Regular exercise

40

63

1.0

No regular exercise

30

71

1.2

  1. Perform the relevant test of hypotheses at the 1% level of significance.

  2. Compute the observed significance of the test.

Ex 19. Children in two elementary school classrooms were given two versions of the same test, but with the order of questions arranged from easier to more difficult in Version A and in reverse order in Version B. Randomly selected students from each class were given Version A and the rest Version B. The results are shown in the table.

n

xˉ\bar{x}

s

Version A

31

83

4.6

Version B

32

78

4.3

  1. Construct the 90% confidence interval for the difference in the means of the populations of all children taking Version A of such a test and of all children taking Version B of such a test.

  2. Test at the 1% level of significance the hypothesis that the A version of the test is easier than the B version (even though the questions are the same).

  3. Compute the observed significance of the test.

Ex 20. The Municipal Transit Authority wants to know if, on weekdays, more passengers ride the northbound blue line train towards the city center that departs at 8:15 a.m. or the one that departs at 8:30 a.m. The following sample statistics are assembled by the Transit Authority.

n

xˉ\bar{x}

s

8:15 a.m. train

30

323

41

8:30 a.m. train

45

356

45

  1. Construct the 90% confidence interval for the difference in the mean number of daily travellers on the 8:15 train and the mean number of daily travellers on the 8:30 train.

  2. Test at the 5% level of significance whether the data provide sufficient evidence to conclude that more passengers ride the 8:30 train.

  3. Compute the observed significance of the test.

Ex 21. In comparing the academic performance of college students who are affiliated with fraternities and those male students who are unaffiliated, a random sample of students was drawn from each of the two populations on a university campus. Summary statistics on the student GPAs are given below.

n

xˉ\bar{x}

s

Fraternity

645

2.90

0.47

Unaffiliated

450

2.88

0.42

Test, at the 5% level of significance, whether the data provide sufficient evidence to conclude that there is a difference in average GPA between the population of fraternity students and the population of unaffiliated male students on this university campus.

Ex 22. In comparing the academic performance of college students who are affiliated with sororities and those female students who are unaffiliated, a random sample of students was drawn from each of the two populations on a university campus. Summary statistics on the student GPAs are given below.

n

xˉ\bar{x}

s

Sorority

330

3.18

0.37

Unaffiliated

550

3.12

0.41

Test, at the 5% level of significance, whether the data provide sufficient evidence to conclude that there is a difference in average GPA between the population of sorority students and the population of unaffiliated female students on this university campus.

Ex 23. The owner of a professional football team believes that the league has become more offense oriented since five years ago. To check his belief, 32 randomly selected games from one year’s schedule were compared to 32 randomly selected games from the schedule five years later. Since more offense produces more points per game, the owner analyzed the following information on points per game (ppg).

n

xˉ\bar{x}

s

ppg previously

32

20.62

4.17

ppg recently

32

22.05

4.01

Test, at the 10% level of significance, whether the data on points per game provide sufficient evidence to conclude that the game has become more offense oriented.

Ex 24. The owner of a professional football team believes that the league has become more offense oriented since five years ago. To check his belief, 32 randomly selected games from one year’s schedule were compared to 32 randomly selected games from the schedule five years later. Since more offense produces more offensive yards per game, the owner analyzed the following information on offensive yards per game (oypg).

n

xˉ\bar{x}

s

oypg previously

32

316

40

oypg recently

32

336

35

Test, at the 10% level of significance, whether the data on offensive yards per game provide sufficient evidence to conclude that the game has become more offense oriented.

3. LARGE DATA SET EXERCISES

Ex 25. Large Data Sets 1A and 1B list the SAT scores for 1,000 randomly selected students. Denote the population of all male students as Population 1 and the population of all female students as Population 2.

  1. Restricting attention to just the males, find n1,x1ˉ,n_1, \bar{x_1}, and s1s_1 . Restricting attention to just the females, find n2,x2ˉ,n_2, \bar{x_2}, and s2s_2.

  2. Let μ1μ_1 denote the mean SAT score for all males and μ2μ_2 the mean SAT score for all females. Use the results of part (a) to construct a 90% confidence interval for the difference (μ1−μ2).(μ_1−μ_2).

  3. Test, at the 5% level of significance, the hypothesis that the mean SAT scores among males exceeds that of females.

Ex 26. Large Data Sets 1A and 1B list the GPAs for 1,000 randomly selected students. Denote the population of all male students as Population 1 and the population of all female students as Population 2.

  1. Restricting attention to just the males, find n1,x1ˉ,n_1, \bar{x_1}, and s1s_1 . Restricting attention to just the females, find n2,x2ˉ,n_2, \bar{x_2}, and s2s_2.

  2. Let μ1μ_1 denote the mean SAT score for all males and μ2μ_2 the mean SAT score for all females. Use the results of part (a) to construct a 95% confidence interval for the difference (μ1−μ2).(μ_1−μ_2).

  3. Test, at the 10% level of significance, the hypothesis that the mean GPAs among males and females differ.

Ex 27. Large Data Sets 7A and 7B list the survival times for 65 male and 75 female laboratory mice with thymic leukemia. Denote the population of all such male mice as Population 1 and the population of all such female mice as Population 2.

  1. Restricting attention to just the males, find n1,x1ˉ,n_1, \bar{x_1}, and s1s_1 . Restricting attention to just the females, find n2,x2ˉ,n_2, \bar{x_2}, and s2s_2.

  2. Let μ1μ_1 denote the mean SAT score for all males and μ2μ_2 the mean SAT score for all females. Use the results of part (a) to construct a 99% confidence interval for the difference (μ1−μ2).(μ_1−μ_2).

  3. Test, at the 1% level of significance, the hypothesis that the mean survival time for males exceeds that for females by more than 182 days (half a year).

  4. Compute the observed significance of the test in part (c).

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