Sample Size for Estimating a Population Proportion
The confidence interval for a population proportion is given by:
 ± z*
The margin of error is
 z* .
Let M be the desired maximum margin of error. Then,
 M ≤ z*
Solving for n,
But we do not have a value of until we collect data, so we need a way to estimate . Let P* = estimated value of . Then
There are two ways to choose a value of P*:
 Use a previous determined value of . That is, you may already have an idea, based on historical data, about what the value should be close to.
 Use P* = 0.5. A result from calculus tells us that the expression
achieves its maximum value when P* = 0.5. Thus, n will be at its maximum if P* = 0.5. If P* = 0.5, the formula for n can more easily be expressed as
.
It is in your interest to choose the smallest value of n that will match your goals, so any value of P* < 0.5 would be preferable if you have some justification for it.
example: Historically, about 60% of a company's products are purchased by people who have purchased products from the company previously. The company is preparing to introduce a new product and wants to generate a 95% confidence interval for the proportion of its current customers who will purchase the new product. They want to be accurate within 3%. How many customers do they need to sample?
solution: Based on historical data, choose P* = 0.6. Then
.
The company needs to sample 1025 customers. Had it not had the historical data, it would have had to use P* = 0.5.
If P* = 0.5, n ≥ = 1067.1. You need a sample of at least 1068 customers. By using P* = 0.6, the company was able to sample 43 fewer customers.
Practice problems for these concepts can be found at:
 Confidence Intervals and Introduction to Inference Multiple Choice Practice Problems for AP Statistics
 Confidence Intervals and Introduction to Inference Free Response Practice Problems for AP Statistics
 Confidence Intervals and Introduction to Inference Review Problems for AP Statistics
 Confidence Intervals and Introduction to Inference Rapid Review for AP Statistics
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