Comparing Two Proportions BPS 7e Chapter 23 © 2015 W. H. Freeman and Company

Two-Sample Problems When comparing proportions from two populations, we do inference about: p1 – p2. p̂1 – p̂2. p1 + p2. p̂1 + p̂2.

Two-Sample Problems (answer) When comparing proportions from two populations, we do inference about: p1 – p2. p̂1 – p̂2. p1 + p2. p̂1 + p̂2.

Sampling Distribution of p^1 – p^2 In an SRS of 103 male and 100 female college students, 61 males and 54 females said that they usually speed on Interstate highways by 10 mph or more. What is p̂1 – p̂2? 0.592 – 0.54 0.61 – 0.54 0.61 – 0.524 0.591 – 0.524

Sampling Distribution of p^1 – p^2 (answer) In an SRS of 103 male and 100 female college students, 61 males and 54 females said that they usually speed on Interstate highways by 10 mph or more. What is p̂1 – p̂2? 0.592 – 0.54 0.61 – 0.54 0.61 – 0.524 0.591 – 0.524

Sampling Distribution of p^1 – p^2 What is the shape of the sampling distribution of p̂1 – p̂2 when the data are obtained as large simple random samples? exactly Normal approximately Normal right-skewed left-skewed

Sampling Distribution of p^1 – p^2 (answer) What is the shape of the sampling distribution of p̂1 – p̂2 when the data are obtained as large simple random samples? exactly Normal approximately Normal right-skewed left-skewed

Sampling Distribution Ten percent of men and 8 percent of women are left-handed. For a sample of 100 men and 200 women, the mean of the sampling distribution of p̂1 – p̂2 is ____________ and the standard deviation of p̂1 – p̂2 is _____________.

Sampling Distribution (answer) Ten percent of men and 8 percent of women are left-handed. For a sample of 100 men and 200 women, the mean of the sampling distribution of p̂1 – p̂2 is ____________ and the standard deviation of p̂1 – p̂2 is _____________.

Sampling Distribution Ten percent of men and 8 percent of women are left-handed. For a sample of 100 men and 200 women, what is the probability that p̂1 – p̂2 would be less than or equal to 5.5%? (Hint: The mean of p̂1 – p̂2 is .02 and the standard deviation of p̂1 – p̂2 is .035.) 16% 34% 68% 84% 97.5%

Sampling Distribution (answer) Ten percent of men and 8 percent of women are left-handed. For a sample of 100 men and 200 women, what is the probability that p̂1 – p̂2 would be less than or equal to 5.5%? (Hint: The mean of p̂1 – p̂2 is .02 and the standard deviation of p̂1 – p̂2 is .035.) 16% 34% 68% 84% 97.5%

Large-Sample Confidence Interval A confidence interval for p̂1 – p̂2 gives a set of reasonable values for the: level of confidence. pooled sample proportion. pooled population proportion. difference between p1 and p2. sum of p1 and p2.

Large-Sample Confidence Interval (answer) A confidence interval for p̂1 – p̂2 gives a set of reasonable values for the: level of confidence. pooled sample proportion. pooled population proportion. difference between p1 and p2. sum of p1 and p2.

Large-Sample Confidence Interval In an SRS of 103 male and 100 female college students, 61 males and 54 females said that they usually speed on Interstate highways by 10 mph or more. What is the standard error of p̂1 – p̂2?

Large-Sample Confidence Interval (answer) In an SRS of 103 male and 100 female college students, 61 males and 54 females said that they usually speed on Interstate highways by 10 mph or more. What is the standard error of p̂1 – p̂2?

Large-Sample Confidence Interval In an SRS of 103 male and 100 female college students, 61 males and 54 females said that they usually speed on Interstate highways by 10 mph or more. Say the standard error of p̂1 – p̂2 is .07. Then, a 95% confidence interval for p1 – p2 would be approximately: 0.052 ± 0.035. 0.052 ± 0.070. 0.052 ± 0.140. 0.052 ± 0.185.

Large-Sample Confidence Interval (answer) In an SRS of 103 male and 100 female college students, 61 males and 54 females said that they usually speed on Interstate highways by 10 mph or more. Say the standard error of p̂1 – p̂2 is .07. Then, a 95% confidence interval for p1 – p2 would be approximately: 0.052 ± 0.035. 0.052 ± 0.070. 0.052 ± 0.140. 0.052 ± 0.185.

Significance Tests We want to test whether proportions from two populations are different from each other. What are the appropriate null and alternative hypotheses? H0: p1 = p2, Ha: p1 ≠ p2 H0: p1 ≠ p2, Ha: p1 = p2 H0: p1 = p2, Ha : p1 > p2 H0: p̂1 = p̂2, Ha: p̂1 ≠ p̂2 H0: p̂1 = p̂2, Ha : p̂1 ≠ p̂2

Significance Tests (answer) We want to test whether proportions from two populations are different from each other. What are the appropriate null and alternative hypotheses? H0: p1 = p2, Ha: p1 ≠ p2 H0: p1 ≠ p2, Ha: p1 = p2 H0 : p1 = p2, Ha: p1 > p2 H0: p̂1 = p̂2, Ha: p̂1 ≠ p̂2

Significance Tests We want to compare two populations and test whether proportion 1 is higher than proportion 2. What are the appropriate null and alternative hypotheses? H0: p1 = p2, Ha: p1 ≠ p2 H0: p1 ≠ p2, Ha: p1 = p2 H0: p1 = p2, Ha: p1 > p2 H0: p̂1 = p̂2, Ha: p̂1 < p̂2 H0: p̂1 = p̂2, Ha: p̂1 ≠ p̂2

Significance Tests (answer) We want to compare two populations and test whether proportion 1 is higher than proportion 2. What are the appropriate null and alternative hypotheses? H0: p1 = p2, Ha: p1 ≠ p2 H0: p1 ≠ p2, Ha: p1 = p2 H0: p1 = p2, Ha: p1 > p2 H0: p̂1 = p̂2, Ha: p̂1 < p̂2 H0: p̂1 = p̂2, Ha: p̂1 ≠ p̂2

Significance Tests When do we use the pooled sample proportion p̂ ? when doing a confidence interval for p1 – p2 when doing a test of H0: p1 = p2

Significance Tests (answer) When do we use the pooled sample proportion p̂ ? when doing a confidence interval for p1 – p2 when doing a test of H0: p1 = p2

Significance Tests In an SRS of 103 male and 100 female college students, 61 males and 54 females said that they usually speed on Interstate highways by 10 mph or more. What is the pooled sample proportion p̂ ?

Significance Tests (answer) In an SRS of 103 male and 100 female college students, 61 males and 54 females said that they usually speed on Interstate highways by 10 mph or more. What is the pooled sample proportion p̂ ?

Significance Tests Why do we use as the test statistic rather than ? The first is easier to compute. They are both fine. The first takes advantage of H0: p1 = p2. The second is grossly wrong.

Significance Tests (answer) Why do we use as the test statistic rather than ? The first is easier to compute. They are both fine. The first takes advantage of H0: p1 = p2. The second is grossly wrong.

H0: pmale = pfemale vs. Ha: pmale > pfemale Significance Tests In an SRS of 103 male and 100 female college students, 61 males and 54 females said that they speed by 10 mph or more. The test statistic for H0: pmale = pfemale vs. Ha: pmale > pfemale is 0.75. Use the 68-95-99.7 rule to guess which p-value is correct. .784 .483 .227 .109

Significance Tests (answer) In an SRS of 103 male and 100 female college students, 61 males and 54 females said that they speed by 10 mph or more. The test statistic for H0: pmale = pfemale vs. Ha: pmale > pfemale is 0.75. Use the 68-95-99.7 rule to guess which p-value is correct. .784 .483 .227 .109

Statistical Significance Suppose the P-value for a hypothesis test is 0.201. Using  = 0.05, what is the appropriate conclusion? Reject the null hypothesis. Reject the alternative hypothesis. Do not reject the null hypothesis. Do not reject the alternative hypothesis.

Statistical Significance (answer) Suppose the P-value for a hypothesis test is 0.201. Using  = 0.05, what is the appropriate conclusion? Reject the null hypothesis. Reject the alternative hypothesis. Do not reject the null hypothesis. Do not reject the alternative hypothesis.

Comparing Proportions True or False: In a pooled sample proportion, two samples are combined to estimate this single p instead of estimating p1 and p2 separately. True False

Comparing Proportions (answer) True or False: In a pooled sample proportion, two samples are combined to estimate this single p instead of estimating p1 and p2 separately. True False

Sample Conditions When the conditions for the large-sample confidence interval are not met, to get a more accurate confidence interval, four imaginary observations can be added—one success and one failure in each sample. Then, the same formula can be used for the confidence interval. This is the_____________. You can use it whenever both samples have _____ or more observations. sampling distribution for proportion; five plus four confidence interval; five plus four confidence interval; three pooled sample proportion; four

Sample Conditions (answer) When the conditions for the large-sample confidence interval are not met, to get a more accurate confidence interval, four imaginary observations can be added—one success and one failure in each sample. Then, the same formula can be used for the confidence interval. This is the_____________. You can use it whenever both samples have _____ or more observations. sampling distribution for proportion; five plus four confidence interval; five plus four confidence interval; three pooled sample proportion; four