9-4. Exercises

1. BASIC

Ex 1. Construct the confidence interval for (p1−p2)(p_1−p_2) for the level of confidence and the data given. (The samples are sufficiently large.)

  1. 90% confidence,

    n1=1670,p1^=0.42n_1=1670, \hat{p_1}=0.42

    n2=900,p2^=0.38n_2=900, \hat{p_2}=0.38

  2. 95% confidence,

    n1=600,p1^=0.84n_1=600, \hat{p_1}=0.84

    n2=420,p2^=0.67n_2=420, \hat{p_2}=0.67

Ex 2. Construct the confidence interval for (p1−p2)(p_1−p_2) for the level of confidence and the data given. (The samples are sufficiently large.)

  1. 98% confidence,

    n1=750,p1^=0.64n_1=750, \hat{p_1}=0.64

    n2=800,p2^=0.51n_2=800, \hat{p_2}=0.51

  2. 99.5% confidence,

    n1=250,p1^=0.78n_1=250, \hat{p_1}=0.78

    n2=250,p2^=0.51n_2=250, \hat{p_2}=0.51

Ex 3. Construct the confidence interval for (p1−p2)(p_1−p_2) for the level of confidence and the data given. (The samples are sufficiently large.)

  1. 80% confidence,

    n1=300,p1^=0.255n_1=300, \hat{p_1}=0.255

    n2=400,p2^=0.193n_2=400, \hat{p_2}=0.193

  2. 95% confidence,

    n1=3500,p1^=0.147n_1=3500, \hat{p_1}=0.147

    n2=3750,p2^=0.131n_2=3750, \hat{p_2}=0.131

Ex 4. Construct the confidence interval for (p1−p2)(p_1−p_2) for the level of confidence and the data given. (The samples are sufficiently large.)

  1. 99% confidence,

    n1=2250,p1^=0.915n_1=2250, \hat{p_1}=0.915

    n2=2520,p2^=0.858n_2=2520, \hat{p_2}=0.858

  2. 95% confidence,

    n1=120,p1^=0.650n_1=120, \hat{p_1}=0.650

    n2=200,p2^=0.505n_2=200, \hat{p_2}=0.505

Ex 5. Perform the test of hypotheses indicated, using the data given. Use the critical value approach. Compute the p-value of the test as well. (The samples are sufficiently large.)

  1. Test H0:p1−p2=0 vs. Ha:p1−p2>0,@α=0.10,H_0:p1−p2=0 \space vs. \space H_a:p1−p2>0, @ α=0.10,

    n1=1200,p1^=0.42n_1=1200, \hat{p_1}=0.42

    n2=1200,p2^=0.40n_2=1200, \hat{p_2}=0.40

  2. Test H0:p1−p2=0 vs. Ha:p1−p2≠0,@α=0.05,H_0:p1−p2=0 \space vs. \space H_a:p1−p2\ne 0, @ α=0.05,

    n1=550,p1^=0.61n_1=550, \hat{p_1}=0.61

    n2=600,p2^=0.67n_2=600, \hat{p_2}=0.67

Ex 6. Perform the test of hypotheses indicated, using the data given. Use the critical value approach. Compute the p-value of the test as well. (The samples are sufficiently large.)

  1. Test H0:p1−p2=0.05 vs. Ha:p1−p2>0.05,@α=0.05,H_0:p1−p2=0.05 \space vs. \space H_a:p1−p2>0.05, @ α=0.05,

    n1=1100,p1^=0.57n_1=1100, \hat{p_1}=0.57

    n2=1100,p2^=0.48n_2=1100, \hat{p_2}=0.48

  2. Test H0:p1−p2=0 vs. Ha:p1−p2≠0,@α=0.05,H_0:p1−p2=0 \space vs. \space H_a:p1−p2\ne 0, @ α=0.05,

    n1=800,p1^=0.39n_1=800, \hat{p_1}=0.39

    n2=900,p2^=0.43n_2=900, \hat{p_2}=0.43

Ex 7. Perform the test of hypotheses indicated, using the data given. Use the critical value approach. Compute the p-value of the test as well. (The samples are sufficiently large.)

  1. Test H0:p1−p2=0.25 vs. Ha:p1−p2<0.25,@α=0.005,H_0:p1−p2=0.25 \space vs. \space H_a:p1−p2<0.25, @ α=0.005,

    n1=1400,p1^=0.57n_1=1400, \hat{p_1}=0.57

    n2=1200,p2^=0.37n_2=1200, \hat{p_2}=0.37

  2. Test H0:p1−p2=0.16 vs. Ha:p1−p2≠0.16,@α=0.02,H_0:p1−p2=0.16 \space vs. \space H_a:p1−p2\ne 0.16, @ α=0.02,

    n1=750,p1^=0.43n_1=750, \hat{p_1}=0.43

    n2=600,p2^=0.22n_2=600, \hat{p_2}=0.22

Ex 8. Perform the test of hypotheses indicated, using the data given. Use the critical value approach. Compute the p-value of the test as well. (The samples are sufficiently large.)

  1. Test H0:p1−p2=0.08 vs. Ha:p1−p2>0.08,@α=0.025,H_0:p1−p2=0.08 \space vs. \space H_a:p1−p2>0.08, @ α=0.025,

    n1=450,p1^=0.67n_1=450, \hat{p_1}=0.67

    n2=200,p2^=0.52n_2=200, \hat{p_2}=0.52

  2. Test H0:p1−p2=0.02 vs. Ha:p1−p2≠0.02,@α=0.001,H_0:p1−p2=0.02 \space vs. \space H_a:p1−p2 \ne 0.02, @ α=0.001,

    n1=2700,p1^=0.837n_1=2700, \hat{p_1}=0.837

    n2=2900,p2^=0.854n_2=2900, \hat{p_2}=0.854

Ex 9. Perform the test of hypotheses indicated, using the data given. Use the p-value approach. (The samples are sufficiently large.)

  1. Test H0:p1−p2=0 vs. Ha:p1−p2<0,@α=0.005,H_0:p1−p2=0 \space vs. \space H_a:p1−p2<0, @ α=0.005,

    n1=1100,p1^=0.22n_1=1100, \hat{p_1}=0.22

    n2=1300,p2^=0.27n_2=1300, \hat{p_2}=0.27

  2. Test H0:p1−p2=0 vs. Ha:p1−p2≠0,@α=0.01,H_0:p1−p2=0 \space vs. \space H_a:p1−p2\ne0, @ α=0.01,

    n1=650,p1^=0.35n_1=650, \hat{p_1}=0.35

    n2=650,p2^=0.41n_2=650, \hat{p_2}=0.41

Ex 10. Perform the test of hypotheses indicated, using the data given. Use the p-value approach. (The samples are sufficiently large.)

  1. Test H0:p1−p2=0.15 vs. Ha:p1−p2>0.15,@α=0.10,H_0:p1−p2=0.15 \space vs. \space H_a:p1−p2>0.15, @ α=0.10,

    n1=950,p1^=0.41n_1=950, \hat{p_1}=0.41

    n2=500,p2^=0.23n_2=500, \hat{p_2}=0.23

  2. Test H0:p1−p2=0.10 vs. Ha:p1−p2≠0.10,@α=0.10,H_0:p1−p2=0.10 \space vs. \space H_a:p1−p2 \ne 0.10, @ α=0.10,

    n1=220,p1^=0.92n_1=220, \hat{p_1}=0.92

    n2=160,p2^=0.78n_2=160, \hat{p_2}=0.78

Ex 11. Perform the test of hypotheses indicated, using the data given. Use the p-value approach. (The samples are sufficiently large.)

  1. Test H0:p1−p2=0.22 vs. Ha:p1−p2>0.22,@α=0.05,H_0:p1−p2=0.22 \space vs. \space H_a:p1−p2>0.22, @ α=0.05,

    n1=90,p1^=0.72n_1=90, \hat{p_1}=0.72

    n2=75,p2^=0.40n_2=75, \hat{p_2}=0.40

  2. Test H0:p1−p2=0.37 vs. Ha:p1−p2≠0.37,@α=0.02,H_0:p1−p2=0.37 \space vs. \space H_a:p1−p2 \ne 0.37, @ α=0.02,

    n1=425,p1^=0.772n_1=425, \hat{p_1}=0.772

    n2=425,p2^=0.331n_2=425, \hat{p_2}=0.331

Ex 12. Perform the test of hypotheses indicated, using the data given. Use the p-value approach. (The samples are sufficiently large.)

  1. Test H0:p1−p2=0.50 vs. Ha:p1−p2<0.50,@α=0.10,H_0:p1−p2=0.50 \space vs. \space H_a:p1−p2<0.50, @ α=0.10,

    n1=40,p1^=0.65n_1=40, \hat{p_1}=0.65

    n2=55,p2^=0.24n_2=55, \hat{p_2}=0.24

  2. Test H0:p1−p2=0.30 vs. Ha:p1−p2≠0.30,@α=0.10,H_0:p1−p2=0.30 \space vs. \space H_a:p1−p2\ne 0.30, @ α=0.10,

    n1=7500,p1^=0.664n_1=7500, \hat{p_1}=0.664

    n2=1000,p2^=0.319n_2=1000, \hat{p_2}=0.319

2. APPLICATIONS

In all the remaining exercises the samples are sufficiently large (so this need not be checked).

Ex 13. Voters in a particular city who identify themselves with one or the other of two political parties were randomly selected and asked if they favor a proposal to allow citizens with proper license to carry a concealed handgun in city parks. The results are:

Party A

Party B

Sample size, n

150

200

Number in favor, x

90

140

  1. Give a point estimate for the difference in the proportion of all members of Party A and all members of Party B who favor the proposal.

  2. Construct the 95% confidence interval for the difference, based on these data.

  3. Test, at the 5% level of significance, the hypothesis that the proportion of all members of Party A who favor the proposal is less than the proportion of all members of Party B who do.

  4. Compute the p-value of the test.

Ex 14. To investigate a possible relation between gender and handedness, a random sample of 320 adults was taken, with the following results:

Men

Women

Sample size, n

168

152

Number of left-handed, x

24

9

  1. Give a point estimate for the difference in the proportion of all men who are left-handed and the proportion of all women who are left-handed.

  2. Construct the 95% confidence interval for the difference, based on these data.

  3. Test, at the 5% level of significance, the hypothesis that the proportion of men who are left-handed is greater than the proportion of women who are.

  4. Compute the p-value of the test.

Ex 15. A local school board member randomly sampled private and public high school teachers in his district to compare the proportions of National Board Certified (NBC) teachers in the faculty. The results were:

Private Schools

Public Schools

Sample size, n

80

520

Proportion of NBC teachers, p^\hat{p}

0.175

0.150

  1. Give a point estimate for the difference in the proportion of all teachers in area public schools and the proportion of all teachers in private schools who are National Board Certified.

  2. Construct the 90% confidence interval for the difference, based on these data.

  3. Test, at the 10% level of significance, the hypothesis that the proportion of all public school teachers who are National Board Certified is less than the proportion of private school teachers who are.

  4. Compute the p-value of the test.

Ex 16. In professional basketball games, the fans of the home team always try to distract free throw shooters on the visiting team. To investigate whether this tactic is actually effective, the free throw statistics of a professional basketball player with a high free throw percentage were examined. During the entire last season, this player had 656 free throws, 420 in home games and 236 in away games. The results are summarized below.

Home

Away

Sample size, n

420

236

Free throw percent, p^\hat{p}

81.5%

78.8%

  1. Give a point estimate for the difference in the proportion of free throws made at home and away.

  2. Construct the 90% confidence interval for the difference, based on these data.

  3. Test, at the 10% level of significance, the hypothesis that there exists a home advantage in free throws.

  4. Compute the p-value of the test.

Ex 17. Randomly selected middle-aged people in both China and the United States were asked if they believed that adults have an obligation to financially support their aged parents. The results are summarized below.

China

USA

Sample size, n

1300

150

Number of yes, x

1170

110

Test, at the 1% level of significance, whether the data provide sufficient evidence to conclude that there exists a cultural difference in attitude regarding this question.

Ex 18. A manufacturer of walk-behind push mowers receives refurbished small engines from two new suppliers, A and B. It is not uncommon that some of the refurbished engines need to be lightly serviced before they can be fitted into mowers. The mower manufacturer recently received 100 engines from each supplier. In the shipment from A, 13 needed further service. In the shipment from B, 10 needed further service. Test, at the 10% level of significance, whether the data provide sufficient evidence to conclude that there exists a difference in the proportions of engines from the two suppliers needing service.

3. LARGE DATA SET EXERCISES

Ex 19. Large Data Sets 6A and 6B record results of a random survey of 200 voters in each of two regions, in which they were asked to express whether they prefer Candidate A for a U.S. Senate seat or prefer some other candidate. Let the population of all voters in region 1 be denoted Population 1 and the population of all voters in region 2 be denoted Population 2. Let p1p_1 be the proportion of voters in Population 1 who prefer Candidate A, and p2p_2 the proportion in Population 2 who do.

  1. Find the relevant sample proportions p1^\hat{p_1} and p2^.\hat{p_2}.

  2. Construct a point estimate for p1−p2.p_1−p_2.

  3. Construct a 95% confidence interval for p1−p2.p_1−p_2.

  4. Test, at the 5% level of significance, the hypothesis that the same proportion of voters in the two regions favor Candidate A, against the alternative that a larger proportion in Population 2 do.

Ex 20. Large Data Set 11 records the results of samples of real estate sales in a certain region in the year 2008 (lines 2 through 536) and in the year 2010 (lines 537 through 1106). Foreclosure sales are identified with a 1 in the second column. Let all real estate sales in the region in 2008 be Population 1 and all real estate sales in the region in 2010 be Population 2.

  1. Use the sample data to construct point estimates p1^\hat{p_1} and p2^\hat{p_2} of the proportions p1p_1 and p2p_2 of all real estate sales in this region in 2008 and 2010 that were foreclosure sales. Construct a point estimate of p1−p2.p_1−p_2.

  2. Use the sample data to construct a 90% confidence for p1−p2.p_1−p_2.

  3. Test, at the 10% level of significance, the hypothesis that the proportion of real estate sales in the region in 2010 that were foreclosure sales was greater than the proportion of real estate sales in the region in 2008 that were foreclosure sales. (The default is that the proportions were the same.)

Previous9-4. Comparison of Two Population ProportionsNext9-5. Sample Size Considerations

Last updated