Section 8.2 Estimating a Population Proportion. ACTIVITY The beads. Conditions for Estimating p

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1 Section 8.2 Estimating a Population Proportion ACTIVITY The beads 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 How can we use this information to find a confidence interval for p? If the sample size is large enough that both np and nq are at least 10, the sampling distribution of is approximately Normal. The mean of the sampling distribution of is p. The standard deviation of the sampling distribution of is. In practice, we do not know the value of p. If we did, we would not need to construct a confidence interval for it! In large samples, will be close to p, so we will replace p with in checking the Normal condition. The Beads (Checking conditions) Mr. Vignolini s class wants to construct a confidence interval for the proportion p of red beads in the container. Recall that the class s sample had 107 red beads and 144 white beads. PROBLEM: Check that the conditions for constructing a confidence interval for p are met. 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. 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.

2 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: The sample proportion is the statistic we use to estimate p. When the Independent condition is met, the standard deviation of the sampling distribution of is Since we don t know p, we replace it with the sample proportion. This gives us the standard error (SE) of the sample proportion: DEFINITION: Standard error When the standard deviation of a statistic is estimated from data, the result is called the standard error of the statistic. Finding a Critical Value How do we find the critical value for our confidence interval? 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 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. 80% Confidence (Finding a critical value) PROBLEM: Use Table A to find the critical value z* for an 80% confidence interval. Assume that the Normal condition is met.

3 One-Sample z Interval for a Population Proportion Once we find the critical value z*, our confidence interval for the population proportion p is One-Sample z Interval for 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. The Beads (Calculating a confidence interval for p) PROBLEM: Mr. Vignolini s class took an SRS of beads from the container and got 107 red beads and 144 white beads. (a) Calculate and interpret a 90% confidence interval for p. (b) Mr. Vignolini claims that exactly half of the beads in the container are red. Use your result from (a) to comment on this claim. 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. 2. Check 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. 4. Interpret the interval in context.

4 Putting It All Together: The Four-Step Process We can use the familiar four-step process whenever a problem asks us to construct and interpret a confidence interval. State: Confidence Intervals: A Four-Step Process Plan: Do: Conclude: Tees Say Sex Can Wait (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. STATE: PLAN: DO: CONCLUDE: Remember that the margin of error in a confidence interval includes only sampling variability. Confidence Interval for a Population Proportion (Calculator)

5 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. 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. Because the margin of error involves the sample proportion, we have to guess the latter value when choosing n. There are two ways to do this: Use a guess for based on past experience or a pilot study Use as the guess. ME is largest when Sample Size for Desired Margin of Error 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: where is a guessed value for the sample proportion. The margin of error will always be less than or equal to ME if you take the guess to be 0.5. Customer Satisfaction (Determining sample size) 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. She decides that she wants the estimate to be within 3% (0.03) at a 95% confidence level. How large a sample is needed? PROBLEM: Determine the sample size needed to estimate p within 0.03 with 95% confidence. 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 level. 2. What if the company president demands 99% confidence instead? Determine how this would affect your answer to Question 1.