# AP Statistics Section 10.3 CI for a Population Proportion.

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AP Statistics Section 10.3 CI for a Population Proportion

What proportion of U.S. adults are unemployed right now? What proportion of teenagers have a computer with internet access in their bedrooms? What proportion of college students pray on a daily basis?

In these situations, we are interested in the unknown proportion p, of a population. For convenience, call the outcome we are looking for a _________.

Reviewing the sampling distribution of from section 9.2:

Center: the mean is ____. That is, the sample proportion is an unbiased estimator of the population proportion p. Spread: The standard deviation of is _____________, provided that the population is at least 10 times as large as the sample. Shape: The distribution of is approximately Normal if the sample size is large enough that both __________ and ______________.

In practice, we don’t know the value of p. So…… Shape: In large samples, will be close to p. Therefore, we replace p with in determining the values of np and n(1 – p). Spread: We replace p with in the formula for standard deviation. This quantity is the standard error of __________________ of. SE =

Conditions for inference about a proportion SRS: the data are an SRS from the population of interest. Normality of : For a CI, n must be large enough that both _________ and _______________. Independent: Individual observations are independent. When sampling without replacement, the population is at least 10 times as large as the sample.

Example: Alcohol abuse is a concern on college campuses. Researchers defined “frequent binge drinking” as having five or more drinks in a row three or more times in the last two weeks. The Harvard School of Public Health sent surveys to a random sample of undergraduate students at 120 colleges and universities. Of the 10,904 students who responded, 2,486 were frequent binge drinkers. Calculate a 99% confidence interval for the proportion p of all college undergraduates who admit to frequent binge drinking.

Parameter:

Conditions SRS: Normality of : Independence:

Calculations

Interpretation

Note: The margin of error in this confidence interval includes only random sampling error! There are other possible sources of error not taken into account, such as?

In planning a study, we may want to choose a sample size that will allow us to estimate the parameter within a given margin of error. Earlier, we did this for a population mean. The method is similar for estimating a population proportion.

The margin of error in the confidence interval for p is… If we want a given margin error, we consider the inequality and solve for n.

T o find our necessary sample size, n, we need to know,, the sample proportion of successes. We will need to “guess” this value. Call our guess p *. Here are two ways to get p*. 1. Use a guess for p* based on a pilot study or on past experience with similar studies. 2. Use p* = _____ as the guess.

Note: Using p* =.5, will give us the largest possible value of n. Thus, you are guaranteed of having a large enough sample size. However, sampling costs money, so if you have value for p* other than.5, you would prefer to use it.

Example: A company has received complaints about its customer service. They 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 “very satisfied” or “satisfied”. She decides that she wants the estimate to be within 3% and have a 95% confidence level.