9.3 Hypothesis Tests for Two Proportions

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CCSS.HSS.IC.B.4

• We have 2 independent random samples.

• The sample size is less than 10% of the size of the population for each group.

• The expected number of successes and failures are at least 10 for each group.

• All of the conditions are met.

• More than one of the conditions are not met.

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CCSS.HSS.IC.B.4

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CCSS.HSS.IC.B.4

• greater than 0.10.

• between 0.05 and 0.10.

• between 0.01 and 0.05.

• between 0.001 and 0.01.

• below 0.001.

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CCSS.HSS.MD.A.2

Phoebe has a theory that older students at her high school are more likely to bring a bag lunch than younger students, because they have grown tired of cafeteria food. She takes a random sample of sophomores and a random sample of seniors and determines the proportion of each sample that bring a bag lunch. Letting p1 = proportion of all sophomores who bring a bag lunch, and p2 = proportion of all seniors who bring a bag lunch, Phoebe tests the hypotheses H0 : p1 – p2 = 0, Ha : p1 – p2 < 0 at the a = 0.05 level. The test statistic is –1.48. Which of the following is closest to the appropriate P-value for the test?

• 0.0694

• 0.0808

• 0.1388

• 0.8612

• 0.9306

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CCSS.HSS.IC.B.4

• 0.0446

• 0.0892

• 0.1423

• 0.1449

• 0.2846

• If the on-time rates for the two airlines are equal, there is a 0.0117 probability of getting samples with a difference as far or farther from zero as these samples.

• If the on-time rates for the two airlines are not equal, the probability of getting samples with a difference as far or farther from zero as these samples is 0.9883.

• The probability of making a Type I error is 0.0117.

• The probability of making a Type II error is 0.0117.

• The probability that H0 is true is 0.0117.

In a large Midwestern university an SRS of 100 entering freshmen in 2014 found that 20 finished in the bottom third of their high school class. In 2016, an SRS of 100 entering freshmen found that 10 finished in the bottom third of their high school class. In both years, the entering freshman class had roughly 6000 students. Let p1 be the proportion of all entering freshmen in 2014 who graduated in the bottom third of their high school class and let p2 be the proportion of all entering freshmen in 2016 who graduated in the bottom third of their high school class. Is there evidence that the proportion of freshmen who graduated in the bottom third of their high school class in 2016 has been reduced, compared to the proportion in 2014? You test the hypotheses H0 : p1 – p2 = 0, Ha : p1 – p2 > 0 using a = 0.05. The test statistic is 1.980. Which of the following is the appropriate P-value and conclusion for your test?

• P-value = 0.047; fail to reject H0; we do not have convincing evidence that the proportion who graduated in the bottom third of their class has been reduced.

• P-value = 0.047; accept Ha; there is convincing evidence that the proportion who graduated in the bottom third of their class has been reduced.

• P-value = 0.024; fail to reject H0; we do not have convincing evidence that the proportion who graduated in the bottom third of their class has been reduced.

• P-value = 0.024; reject H0; we have convincing evidence that the proportion who graduated in the bottom third of their class has been reduced.

• P-value = 0.024; fail to reject H0; we have convincing evidence that the proportion who graduated in the bottom third of their class has not changed.

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CCSS.HSS.MD.A.2

An SRS of 45 male employees at a large company found that 36 felt that the company was supportive of female employees. An independent SRS of 40 female employees found that 24 felt that the company was supportive of female employees. Let p1 = the proportion of all male employees at the company who hold this opinion and p2 = the proportion of all female employees at the company who hold this opinion. We want to test the hypotheses H0 : p1 – p2 = 0, Ha : p1 – p2 > 0. The P-value for this test is 0.0217. Which of the following is the correct conclusion using a = 0.01?

• Reject H0. We have convincing evidence that the proportion of male employees who feel that the company is supportive of female employees is higher than the proportion of women who feel this way.

• Reject H0. We do not have convincing evidence that the proportion of male employees who feel that the company is supportive of female employees is higher than the proportion of women who feel this way.

• Accept H0. We do not have convincing evidence that the proportion of male employees who feel that the company is supportive of female employees is higher than the proportion of women who feel this way.

• Accept Ha. We do not have convincing evidence that the proportion of male employees who feel that the company is supportive of female employees is higher than the proportion of women who feel this way.

• Fail to reject H0. We do not have convincing evidence that the proportion of male employees who feel that the company is supportive of female employees is higher than the proportion of women who feel this way.

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CCSS.HSS.IC.B.4