Section 9.2b Tests about a Population Proportion

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1 Two-Tailed Tests The basketball player problem and the potato problem were both examples of a single sided or one tailed test of significance. The next problem is a situation that involves a two-sided (or two tailed) test of significance. 1. According to the Centers for Disease Control and Prevention (CDC) Web site, 50% of high school students have never smoked a cigarette. James wonders whether this national result holds true in his large, urban high school. For his AP Statistics class project, James surveys an SRS of 150 students from his school. He gets responses from all 150 students, and 90 say that they have never smoked a cigarette. a. What should James conclude? Give appropriate evidence to support your answer. Assume Page 1 of 14

2 Using a Confidence Interval b. When the alternative hypothesis yields a two-sided test like in this cigarette problem we can use a confidence interval to conduct the test of significance. i. If 0.05 then any sample statistic more than 1.96 standard deviations from the hypothesized value would cause us to reject the null hypothesis. What confidence level does this correspond to? ii. The randomization and independence conditions for constructing the confidence interval are the same and we have checked those already. However the check for normality and the standard error calculation change slightly. When we construct a confidence interval we are not assuming a value for the population parameter so we use ˆp instead of p 0 to verify normality and to calculate the standard error. Verify the normality condition using ˆp. Page 2 of 14

3 iii. Construct the confidence interval (be sure to use ˆp instead of 0 p ). Based on your confidence interval would you reject or fail to reject the null hypothesis. Discuss your reasoning. Page 3 of 14

4 CHECK YOUR UNDERSTANDING Section 9.2b Tests about a Population Proportion 2. According to the National Institute for Occupational Safety and Health, job stress poses a major threat to the health of workers. In a national survey of restaurant employees, 75% said that work stress had a negative impact on their personal lives. Managers of a large restaurant chain wonder whether this national result holds for their employees. A random sample of 100 employees finds that 68 answer Yes when asked, Does work stress have a negative impact on your personal life? Is this good reason to think that the proportion of all employees in this chain who would say Yes differs from the national proportion of 0.75? a. Support your answer with a significance test. Assume an alpha level of 5% (This is a typical alpha level) Page 4 of 14

5 b. Conduct this test using an appropriate level confidence interval. Make sure you recheck the conditions for normality and calculate the standard error using ˆp instead of p 0. Page 5 of 14

6 We have seen in the preceding problems that there is a connection between confidence intervals and two-sided tests. On the AP exam, it is acceptable for you to use a confidence interval for a test of significance that is twosided. There is a connection between one sided tests and confidence intervals but it s more complicated so don t use a confidence interval in place of a one sided test. Sometimes the significance test is conducted for you and the results are presented as computer output (usually from Minitab). The following problem illustrates this idea. 3. What proportion of students are willing to report cheating by other students? A student project put this question to an SRS of 172 undergraduates at a large university: You witness two students cheating on a quiz. Do you go to the professor? The Minitab output below shows the results of a significance test and a 95% confidence interval based on the survey data. a. Define the parameter of interest. b. From the computer output state the null and the alternative hypothesis. c. Interpret the P-value in context. d. Do these data provide convincing evidence at the 0.05 level of significance that the actual population proportion differs from 0.15? Justify your answer with appropriate evidence. Page 6 of 14

7 e. Explain how the confidence interval supports the same conclusion. CHECK YOUR UNDERSTANDING The figure below shows Minitab output from a significance test and confidence interval for the restaurant worker data from the previous check your understanding in this investigation. Verify this confidence interval and explain how the confidence interval is consistent with but gives more information than the test. Page 7 of 14

8 Two-Tailed Tests The basketball player problem and the potato problem were both examples of a single sided or one tailed test of significance. The next problem is a situation that involves a two-sided (or two tailed) test of significance. 1. According to the Centers for Disease Control and Prevention (CDC) Web site, 50% of high school students have never smoked a cigarette. James wonders whether this national result holds true in his large, urban high school. For his AP Statistics class project, James surveys an SRS of 150 students from his school. He gets responses from all 150 students, and 90 say that they have never smoked a cigarette. a. What should James conclude? Give appropriate evidence to support your answer. Assume Page 8 of 14

9 Using a Confidence Interval b. When the alternative hypothesis yields a two-sided test like in this cigarette problem we can use a confidence interval to conduct the test of significance. i. If 0.05 then any sample statistic more than 1.96 standard deviations from the hypothesized value would cause us to reject the null hypothesis. What confidence level does this correspond to? ii. The randomization and independence conditions for constructing the confidence interval are the same and we have checked those already. However the check for normality and the standard error calculation change slightly. When we construct a confidence interval we are not assuming a value for the population parameter so we use ˆp instead of p 0 to verify normality and to calculate the standard error. Verify the normality condition using ˆp. Page 9 of 14

10 iii. Construct the confidence interval (be sure to use ˆp instead of 0 p ). Based on your confidence interval would you reject or fail to reject the null hypothesis. Discuss your reasoning. Page 10 of 14

11 CHECK YOUR UNDERSTANDING Section 9.2b Tests about a Population Proportion 2. According to the National Institute for Occupational Safety and Health, job stress poses a major threat to the health of workers. In a national survey of restaurant employees, 75% said that work stress had a negative impact on their personal lives. Managers of a large restaurant chain wonder whether this national result holds for their employees. A random sample of 100 employees finds that 68 answer Yes when asked, Does work stress have a negative impact on your personal life? Is this good reason to think that the proportion of all employees in this chain who would say Yes differs from the national proportion of 0.75? a. Support your answer with a significance test. Assume an alpha level of 5% (This is a typical alpha level) Page 11 of 14

12 b. Conduct this test using an appropriate level confidence interval. Make sure you recheck the conditions for normality and calculate the standard error using ˆp instead of p 0. Page 12 of 14

13 We have seen in the preceding problems that there is a connection between confidence intervals and two-sided tests. On the AP exam, it is acceptable for you to use a confidence interval for a test of significance that is twosided. There is a connection between one sided tests and confidence intervals but it s more complicated so don t use a confidence interval in place of a one sided test. Sometimes the significance test is conducted for you and the results are presented as computer output (usually from Minitab). The following problem illustrates this idea. 3. What proportion of students are willing to report cheating by other students? A student project put this question to an SRS of 172 undergraduates at a large university: You witness two students cheating on a quiz. Do you go to the professor? The Minitab output below shows the results of a significance test and a 95% confidence interval based on the survey data. a. Define the parameter of interest. b. From the computer output state the null and the alternative hypothesis. c. Interpret the P-value in context. d. Do these data provide convincing evidence at the 0.05 level of significance that the actual population proportion differs from 0.15? Justify your answer with appropriate evidence. Page 13 of 14

14 e. Explain how the confidence interval supports the same conclusion. CHECK YOUR UNDERSTANDING The figure below shows Minitab output from a significance test and confidence interval for the restaurant worker data from the previous check your understanding in this investigation. Verify this confidence interval and explain how the confidence interval is consistent with but gives more information than the test. Page 14 of 14

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