Very frequently asked question in statistical consulting is, how large should the sample size be to estimate a mean?

The answer will depend on the variation associated with the random variable under observation. The statistician could correctly respond, only one item is needed, provided that the standard deviation of the distribution is zero. That is, if

### Example 1

A mathematics department wishes to evaluate a new method of teaching calculus that does mathematics using a computer. At the end of the course, the evaluation will be made on the basis of scores of the participating students on a standard test. Because there is an interest in estimating the mean score *n*, who are to be selected at random from a larger group. So, let find the sample size *n* such that we are fairly confident that

That is, *n*=865 because *n* must be an integer. It is quite likely that it had not been anticipated that as many as 865 students would be needed in this study. If that is the case, the statistician must discuss with those involved in the experiment whether or not the accuracy and the confidence level could be relaxed some. For illustration, rather than requiring *n* must be an integer = 93 is used in practice.

Most likely, the person involved in this project would find this a more reasonable sample size. Of course, any sample size greater than 93 could be used. Then either the length of the confidence interval could be decreased from that of

For example, suppose that the sample characteristics observed are

In general, if we want the *n* is the solution of

That is,

Sometimes

The type of statistic we see most often in newspaper and magazines is an estimate of a proportion *p*. We might, for example, want to know the percentage of the labor force that is unemployed or the percentage of voters favoring a certain candidate. Sometimes extremely important decisions are made on the basis of these estimates. If this is the case, we would most certainly desire short confidence intervals for *p* with large confidence coefficients. We recognize that these conditions will require a large sample size. On the other hand, if the fraction *p* being estimated is not too important, an estimate associated with a longer confidence interval with a smaller confidence coefficients is satisfactory; and thus a smaller sample size can be used.

In general, to find the required sample size to estimate *p*, recall that the point estimate of *p* is

Suppose we want an estimate of *p* that is within *p* with *n*. However, if it is known that *p* is about equal to *n* is the solution of