Elementary Statistics – Confidence Interval for a Population Mean Example

Example 1 (Motivation of the Confidence-Interval Method)

Suppose Governor Doyle is interested in the mean score on the Wechsler Intelligence Scale for Children (WISC) for all children in Wisconsin. He has enlisted our help in estimating this mean.

A random sample of 100 children is given the test. The sample mean is 97.06 (no units). Although it is an unrealistic, for now assume the standard deviation of the population of individual scores is 18 (we’ll later develop inference procedures when the population standard deviation is unknown).

How can we use our one sample mean to estimate the population mean? Create and interpret a 95% confidence interval for the mean score for the entire population of Wisconsin children. (Use the Central Limit Theorem—woo-hoo!)

Example 1 Continued

Find a 90% confidence interval for the mean WISC score for all Wisconsin children. (How does the width of this confidence interval compare with the 95% interval you found previously?)

Example 1 Continued Even More

Now suppose a random sample of 200 children is given the exam, and the sample mean is 97.06. Again, find a 90% confidence interval for the mean WISC score for Wisconsin students. (How does the width of this confidence interval compare with the 90% interval you found previously?)

Example 1 Completely Beaten to Death (Sample Size Determination)

Suppose Governor Doyle wants a 95% confidence interval, but he wants the length of the interval to be no more than 4 points. How large of a sample of students should take the test in order to achieve a margin of error of 2 for a 95% confidence level?