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Sample Size for AP Statistics (page 2)

By — McGraw-Hill Professional
Updated on Feb 4, 2011

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,

    Mz*

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*:

  1. 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.
  2. Use P* = 0.5. A result from calculus tells us that the expression
  3. 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:

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