1. + Chapter 8: Estimating with Confidence Section 8.2 Estimating a Population Proportion

2. + Activity: The Beads Your teacher has a container full of different colored beads. Your goal is to estimate the actual proportion of red beads in the container. Form teams of 3 or 4 students. Determine how to use a cup to get a simple random sample of beadsfrom the container. Each team is to collect one SRS of beads. Determine a point estimate for the unknown population proportion. Find a 90% confidence interval for the parameter p. Consider any conditions that are required for the methods you use. Compare your results with the other teams in the class.

3. + Conditions for Estimating p Suppose one SRS of beads resulted in 107 red beads and 144 beads of another color. The point estimate for the unknown proportion p of red beads in the population would be Estimating a Population Proportion How can we use this information to find a confidence interval for p?

4. + Conditions for Estimating p Check the conditions for estimating p from our sample. Estimating a Population Proportion Random: The class took an SRS of 251 beads from the container. Normal: Both np and n(1 – p) must be greater than 10. Since we don’t know p, we check that The counts of successes (red beads) and failures (non-red) are both ≥ 10. Independent: Since the class sampled without replacement, they need to check the 10% condition. At least 10(251) = 2510 beads need to be in the population. The teacher reveals there are 3000 beads in the container, so the condition is satisfied. Since all three conditions are met, it is safe to construct a confidence interval.

5. + Checkpoint In each of the following settings, check whether the conditions for calculating a confidence interval for the population proportion p are met. 1. An AP Statistics class at a large high school conducts a survey. They ask the first 100 students to arrive at school one morning whether or not they slept at least 8 hours the night before. Only 17 students say “Yes.” Random: Not met. This was a convenience sample. Normal: Met. n = 17 and n (1 – ) = 83 are both at least 10. Independent: Met. A large high school has more than 10 (100) = 1000 students.

6. + Checkpoint 2. A quality control inspector takes a random sample of 25 bags of potato chips from the thousands of bags filled in an hour. Of the bags selected, 3 had too much salt. Random: Met. Normal: Not met. n = 3 is not at least 10. Independent: Met. There are thousands of bags produced per hour, so the sample is less than 10% of the population.

7. + Constructing a Confidence Interval for p We can use the general formula from Section 8.1 to construct a confidence interval for an unknown population proportion p : Estimating a Population Proportion Definition: When the standard deviation of a statistic is estimated from data, the results is called the standard error of the statistic.

8. + Finding a Critical Value How do we find the critical value for our confidence interval? Estimating a Population Proportion If the Normal condition is met, we can use a Normal curve. To find a level C confidence interval, we need to catch the central area C under the standard Normal curve. For example, to find a 95% confidence interval, we use a critical value of 2 based on the 68-95-99.7 rule. Using Table A or a calculator, we can get a more accurate critical value. Note, the critical value z* is actually 1.96 for a 95% confidence level.

9. + Finding a Critical Value Use Table A to find the critical value z* for an 80% confidence interval. Assume that the Normal condition is met. Estimating a Population Proportion Since we want to capture the central 80% of the standard Normal distribution, we leave out 20%, or 10% in each tail. Use InvNorm (0.90, 0, 1) So, the critical value z* for an 80% confidence interval is z* = 1.28. The closest entry is z = – 1.28.

10. + One-Sample z Interval for a Population Proportion Once we find the critical value z*, our confidence interval for the population proportion p is Estimating a Population Proportion Choose an SRS of size n from a large population that contains an unknown proportion p of successes. An approximate level C confidence interval for p is where z* is the critical value for the standard Normal curve with area C between – z* and z*. Use this interval only when the numbers of successes and failures in the sample are both at least 10 and the population is at least 10 times as large as the sample. One-Sample z Interval for a Population Proportion

11. + Calculate and interpret a 90% confidence interval for the proportion of red beads in the container. Your teacher claims 50% of the beads are red.Use your interval to comment on this claim. Estimating a Population Proportion For a 90% confidence level, z* = 1.645 We checked the conditions earlier. sample proportion = 107/251 = 0.426 statistic ± (critical value) (standard deviation of the statistic) We are 90% confident that the interval from 0.375 to 0.477 captures the actual proportion of red beads in the container. Since this interval gives a range of plausible values for p and since 0.5 is not contained in the interval, we have reason to doubt the claim.

12. + Checkpoint Alcohol abuse has been described by college presidents as the number one problem on campus, and it is an important cause of death in young adults. How common is it? A survey of 10,904 randomly selected U.S. college students collected information on drinking behavior and alcohol-related problems. The researchers defined “frequent binge drinking” as having five or more drinks in a row three or more times in the past two weeks. According to this definition, 2486 students were classified as frequent binge drinkers. 1. Identify the population and the parameter of interest. Population: U.S. college students. Parameter: true proportion who are classified as binge drinkers

13. + Checkpoint 2. Check the conditions for constructing a confidence interval for the parameter. 3. Find the critical value for a 99% confidence interval. Show your method. Then calculate the interval. Random: Students were chosen randomly. Normal: 2486 successes and 8418 failures, both are at least 10. Independent: 10,904 is clearly less than 10% of all U.S. college students. ALL CONDITIONS ARE MET! z* = 2.576 (0.218, 0.238)

14. + Checkpoint 4. Interpret the interval in context. We are 99% confident that the interval from 21.8% to 23.8% captures the true proportion of U.S. college students who would be classified as binge drinkers.

15. + The Four-Step Process We can use the familiar four-step process whenever a problem asks us to construct and interpret aconfidence interval. Estimating a Population Proportion State: What parameter do you want to estimate, and at what confidence level? Plan: Identify the appropriate inference method. Check conditions. Do: If the conditions are met, perform calculations. Conclude: Interpret your interval in the context of the problem. Confidence Intervals: A Four-Step Process

16. + Example – Confidence Interval for p The Gallup Youth Survey asked a random sample of 439 U.S. teens aged 13 to 17 whether they thought young people should wait to have sex until marriage. Of the sample, 246 said “Yes.” Construct and interpret a 95% confidence interval for the proportion of all teens who would say “Yes” if asked this question. Use the 4-step process. State: We want to estimate the actual proportion p of all 13 to 17 year olds in the U.S. who would say that young people should wait to have sex until they get married at a 95% confidence interval.

17. Plan: We should use a one-sample z-interval for p if the conditions are satisfied. Random: Gallup survey a random sample of 439 U.S. teens Normal: We check the counts of “successes” and “failures”: Independence: The responses of individual teens need to be independent. Since Gallup is sampling without replacement, we need to check the 10% condition: there are at least 10(439) = 4390 U.S. teens aged 13 to 17.

18. Do: A 95% confidence interval for p is given by: Conclude: We are 95% confident that the interval from 0.514 to 0.606 captures the true proportion of 13 to 17 year olds in the United States who would say that teens should wait until marriage to have sex.

19. + Choosing the Sample Size In planning a study, we may want to choose a sample size that allows us to estimate a population proportion within a given margin of error. Estimating a Population Proportion The margin of error (ME) in the confidence interval for p is z* is the standard Normal critical value for the level of confidence we want. To determine the sample size n that will yield a level C confidence interval for a population proportion p with a maximum margin of error ME, solve the following inequality for n: Sample Size for Desired Margin of Error

20. + Example: Customer Satisfaction A company has received complaints about its customer service. The managers intend to hire a consultant to carry out a survey of customers. Before contacting the consultant, the company president wants some idea of the sample size that she will be required to pay for. One critical question is the degree of satisfaction with the company’s customer service, measured on a five-point scale. The president wants to estimate the proportion p of customers who are satisfied (that is, who choose either “satisfied” or “very satisfied,” the two highest level on the five-point scale). She decides that she wants the estimate to be within 3% (0.03) at a 95% confidence level. How large a sample is needed? Estimating a Population Proportion

21. + Example: Customer Satisfaction Determine the sample size needed to estimate p within 0.03 with 95% confidence. Estimating a Population Proportion The critical value for 95% confidence is z* = 1.96. Since the company president wants a margin of error of no more than 0.03, we need to solve the equation Substitute 0.5 for the sample proportion to find the largest ME possible. We round up to 1068 respondents to ensure the margin of error is no more than 0.03 at 95% confidence.

22. + Checkpoint Refer to the previous example about the company’s customer satisfaction survey. 1. In the company’s prior-year survey, 80% of customers surveyed said they were “satisfied” or “very satisfied.” Using this value as a guess for, find the sample size needed for a margin of error of 3% at a 95% confidence interval.

23. + Checkpoint 2. What if the company demands 99% confidence instead? Determine how this would affect your answer to Question 1.

24. + Section 8.2 Estimating a Population Proportion In this section, we learned that… Summary

25. + Section 8.2 Estimating a Population Proportion In this section, we learned that… When constructing a confidence interval, follow the familiar four-step process: STATE: What parameter do you want to estimate, and at what confidence level? PLAN: Identify the appropriate inference method. Check conditions. DO: If the conditions are met, perform calculations. CONCLUDE: Interpret your interval in the context of the problem. The sample size needed to obtain a confidence interval with approximate margin of error ME for a population proportion involves solving Summary