Comparing Two Proportions BPS 7 e Chapter 23

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

Two-Sample Problems When comparing proportions from two populations, we do inference about: a) b) c) d) p 1 – p 2. p 1 + p 2.

Two-Sample Problems (answer) When comparing proportions from two populations, we do inference about: a) b) c) d) p 1 – p 2. p 1 + p 2.

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 p 1 – p 2? a) b) c) d) 0. 592 – 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 p 1 – p 2? a) b) c) d) 0. 592 – 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 p 1 – p 2 when the data are obtained as large simple random samples? a) b) c) d) 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 p 1 – p 2 when the data are obtained as large simple random samples? a) b) c) d) 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 p 1 – p 2 is ______ and the standard deviation of p 1 – p 2 is _______. a) b) c) d) e)

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 p 1 – p 2 is ______ and the standard deviation of p 1 – p 2 is _______. a) b) c) d) e)

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 p 1 – p 2 would be less than or equal to 5. 5%? (Hint: The mean of p 1 – p 2 is. 02 and the standard deviation of p 1 – p 2 is. 035. ) a) b) c) d) e) 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 p 1 – p 2 would be less than or equal to 5. 5%? (Hint: The mean of p 1 – p 2 is. 02 and the standard deviation of p 1 – p 2 is. 035. ) a) b) c) d) e) 16% 34% 68% 84% 97. 5%

Large-Sample Confidence Interval A confidence interval for p 1 – p 2 gives a set of reasonable values for the: a) b) c) d) e) level of confidence. pooled sample proportion. pooled population proportion. difference between p 1 and p 2. sum of p 1 and p 2.

Large-Sample Confidence Interval (answer) A confidence interval for p 1 – p 2 gives a set of reasonable values for the: a) b) c) d) e) level of confidence. pooled sample proportion. pooled population proportion. difference between p 1 and p 2. sum of p 1 and p 2.

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 p 1 – p 2? a) b) c) d)

Large-Sample Confidence Interval In an SRS of 103 male and 100 female college students, 61 males and (answer) 54 females said that they usually speed on Interstate highways by 10 mph or more. What is the standard error of p 1 – p 2? a) b) c) d)

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 p 1 – p 2 is. 07. Then, a 95% confidence interval for p 1 – p 2 would be approximately: a) b) c) d) 0. 052 ± 0. 035. 0. 052 ± 0. 070. 0. 052 ± 0. 140. 0. 052 ± 0. 185.

Large-Sample Confidence Interval In an SRS of 103 male and 100 female college students, 61 males and (answer) 54 females said that they usually speed on Interstate highways by 10 mph or more. Say the standard error of p 1 – p 2 is. 07. Then, a 95% confidence interval for p 1 – p 2 would be approximately: a) b) c) d) 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? a) b) c) d) e) H 0 : p 1 = p 2 , H a : p 1 ≠ p 2 H 0 : p 1 ≠ p 2 , H a : p 1 = p 2 H 0 : p 1 = p 2 , H a : p 1 > p 2 H 0: p 1 = p 2, Ha: p 1 ≠ p 2 H 0: p 1 = p 2, Ha : p 1 ≠ p 2

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? a) b) c) d) e) H 0 : p 1 = p 2 , Ha : p 1 ≠ p 2 H 0 : p 1 ≠ p 2 , Ha : p 1 = p 2 H 0 : p 1 = p 2 , Ha : p 1 > p 2 H 0: p 1 = p 2, Ha: p 1 ≠ p 2

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? a) b) c) d) e) H 0 : p 1 = p 2 , H a : p 1 ≠ p 2 H 0 : p 1 ≠ p 2 , H a : p 1 = p 2 H 0 : p 1 = p 2 , H a : p 1 > p 2 H 0: p 1 = p 2, Ha: p 1 < p 2 H 0: p 1 = p 2, Ha: p 1 ≠ p 2

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? a) b) c) d) e) H 0 : p 1 = p 2 , H a : p 1 ≠ p 2 H 0 : p 1 ≠ p 2 , H a : p 1 = p 2 H 0 : p 1 = p 2 , H a : p 1 > p 2 H 0: p 1 = p 2, Ha: p 1 < p 2 H 0: p 1 = p 2, Ha: p 1 ≠ p 2

Significance Tests When do we use the pooled sample proportion p ? a) b) when doing a confidence interval for p 1 – p 2 when doing a test of H 0: p 1 = p 2

Significance Tests (answer) When do we use the pooled sample proportion p ? a) b) when doing a confidence interval for p 1 – p 2 when doing a test of H 0: p 1 = p 2

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 p ? a) b) c) d)

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 p ? a) b) c) d)

Significance Tests Why do we use than a) b) c) d) as the test statistic rather ? The first is easier to compute. They are both fine. The first takes advantage of H 0: p 1 = p 2. The second is grossly wrong.

Significance Tests (answer) Why do we use than a) b) c) d) as the test statistic rather ? The first is easier to compute. They are both fine. The first takes advantage of H 0: p 1 = p 2. The second is grossly wrong.

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 H 0: pmale = pfemale vs. Ha: pmale > pfemale is 0. 75. Use the 68 -95 -99. 7 rule to guess which p-value is correct. a) b) c) d) . 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 H 0: pmale = pfemale vs. Ha: pmale > pfemale is 0. 75. Use the 68 -95 -99. 7 rule to guess which p-value is correct. a) b) c) d) . 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? a) b) c) d) 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? a) b) c) d) 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 p 1 and p 2 separately. a) b) 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 p 1 and p 2 separately. a) b) 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. a) b) c) d) sampling distribution for proportion; 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. a) b) c) d) sampling distribution for proportion; five plus four confidence interval; three pooled sample proportion; four