AP Statistics

• Section 10.3: Estimating a Population Proportion

• Objective: To be able to create a confidence interval for a population proportion .

• Recall: • Parameter: p = population proportion• Statistic: = sample proportion• Inference is based on the sampling distribution of .• If certain conditions are met, . However in practice we

do NOT know .• Since is an unbiased estimator of we will estimate with

in our calculations.• Our new standard error for is

• A level C confidence interval for . (1-proportion z-interval).• Conditions:• 1. Data is an SRS.• 2. Normality: n is so larger that and .• 3. Independence: a. Observations are independent.

OR b. When sampling without replacement, population 10n.

General form:

Notice that we are using z critical values instead of t critical values.

Example: Spinning pennies. (pool data together)Create a 95% confidence interval for the true proportion of time a spinning penny falls heads up.

Margin of Error and Choosing a Sample Size:To choose a sample size with a given margin of error, solve the following inequality for n:

where is eithera. ______ which is a conservative value for that will

maximize m. (Use this value as your default)b. could be a value based on prior experiments/studies

or results and may be a value other than the one in a.

Example: What is the minimum sample size required to determine the percentage of time the spinning penny falls heads up with a margin of error of 1%?a. Conservative method

b. Use the historic value given in class to answer again.

Example: A random sample of 71 MHS students revealed that 24 wanted to attend a state funded college or university. Create a 90% confidence interval for the true proportion of MHS students that want to attend a state funded college or university.

If we wanted the interval to have a width of no more than 8%, how many students would we need to sample? Use the conservative method.