Example 8.12 Controlling Confidence Interval Length

| 8.2 | 8.3 | 8.4 | 8.5 | 8.6 | 8.7 | 8.8 | 8.9 | 8.10 | 8.11 | 8.13 | Background Information n Suppose that the fast-food manager from the previous example wants to estimate the proportion of customers who have tried its new sandwich. n It wants a 90% confidence interval for this proportion to have half-length n For example, if the sample proportion turns out to be 0.42, then a 90% confidence interval should be (approximately) or n How many customers need to be surveyed?

| 8.2 | 8.3 | 8.4 | 8.5 | 8.6 | 8.7 | 8.8 | 8.9 | 8.10 | 8.11 | 8.13 | Sample Size Estimation n 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. n The confidence interval for the difference between means uses a t-multiple which should be replaced with the z-multiple. n The confidence intervals for differences between means or proportions requires two sample sizes, one for each sample.

| 8.2 | 8.3 | 8.4 | 8.5 | 8.6 | 8.7 | 8.8 | 8.9 | 8.10 | 8.11 | 8.13 | Sample Size Estimation -- continued n The sample size for a proportion is n Here p est is the estimate of the population proportion p. A conservative value of n can be obtained by using p est = 0.5. This guarantees a confidence interval half- length no greater than B.

| 8.2 | 8.3 | 8.4 | 8.5 | 8.6 | 8.7 | 8.8 | 8.9 | 8.10 | 8.11 | 8.13 | Calculation n Since the manager has no idea what the proportions are, then she can use p est = 0.5 in the equation to obtain a conservative value of n. n The appropriate z-multiple is now because this value cuts off probability 0.05 in each tail of the standard normal distribution. n The formula yields a rounded result of n = 271. n The StatPro add-in can be used to do the same calculation. Using p est = 0.3 gives a result of 228.

| 8.2 | 8.3 | 8.4 | 8.5 | 8.6 | 8.7 | 8.8 | 8.9 | 8.10 | 8.11 | 8.13 | Results n 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. n We also selected a lower confidence level of 90% which gives us less confidence in the result, but it requires a smaller sample size.