## Example

Suppose that the distribution of height of men in a population is normally distributed with mean = i inches and a = 2.5 inches. A sample of 100 men drawn randomly from this population had an average height of 67 inches. Establish a 95% confidence interval for the mean height (= i) in the population as a whole.

As noted, X~ N(i, a2/n), which in this case becomes X~ N(i, 2.52/100). From Table D.1 one can see that

covers 95% of the area under the normal curve. Therefore, the preceding interval provides a 95% confidence interval for i. Plugging in the given values of X, a, and n, we obtain the 95% confidence interval as

In repeated such measurements, intervals thus established will include the true i with 95 percent confidence. A technical point may be noted here. Although we can say that the probability that the random interval [X± 1.96(a/V")] includes i is 95 percent, we cannot say that the probability is 95 percent that the particular interval (66.51, 67.49) includes i. Once this interval is fixed, the probability that it will include i is either 0 or 1. What we can say is that if we construct 100 such intervals, 95 out of the 100 intervals will include the true i; we cannot guarantee that one particular interval will necessarily include i.

898 APPENDIX A: A REVIEW OF SOME STATISTICAL CONCEPTS 