Example 8.12

Controlling Confidence Interval Length

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Background Information

Suppose that the fast-food manager from the previous example wants to estimate the proportion of customers who have tried its new sandwich.

It wants a 90% confidence interval for this proportion to have half-length 0.05.

For example, if the sample proportion turns out to be 0.42, then a 90% confidence interval should be (approximately) 0.42 +or - 0.05.

How many customers need to be surveyed?

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Sample Size Estimation

The sample size estimation for the mean carries over with few changes to other parameters such as the proportion, difference between two means, and the difference between two proportions.

The confidence interval for the difference between means uses a t-multiple which should be replaced with the z-multiple.

The confidence intervals for differences between means or proportions requires two sample sizes, one for each sample.

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Sample Size Estimation -- continued The sample size for a proportion is

Here pest is the estimate of the population proportion p. A conservative value of n can be obtained by using pest = 0.5. This guarantees a confidence interval half-length no greater than B.

271)5.01)(5.0(05.0

645.12

n

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Calculation

Since the manager has “no idea” what the proportions are, then she can use pest = 0.5 in the equation to obtain a conservative value of n.

The appropriate z-multiple is now 1.645 because this value cuts off probability 0.05 in each tail of the standard normal distribution.

The formula yields a rounded result of n = 271.

The StatPro add-in can be used to do the same calculation. Using pest = 0.3 gives a result of 228.

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Results

The calculations indicate that if we have more specific information about the unknown proportion, we can get by with a smaller sample size - in this case 228 instead of 271.

We also selected a lower confidence level of 90% which gives us less confidence in the result, but it requires a smaller sample size.