## Calculating the required sample size

Before collecting sample data it is obviously necessary to know how large the sample size has to be. The required sample size will depend upon two factors:

■ the desired level of precision of the estimate, and

■ the funds available to carry out the survey.

The greater the precision required the larger the sample size needs to be, other things being equal. But a larger sample will obviously cost more to collect and this might conflict with a limited amount of funds being available. There is a trade-off therefore between the two desirable objectives of high precision and low cost. The following example shows how these two objectives conflict.

A firm producing sweets wishes to find out the average amount of pocket money children receive per week. It wants to be 99% confident that the estimate is within 20 pence of the correct value. How large a sample is needed?

The problem is one of estimating a confidence interval, turned on its head. Instead of having the sample information X, s and n, and calculating the confidence interval for ¡¡, the desired width of the confidence interval is given and it is necessary to find the sample size n which will ensure this. The formula for the 99% confidence interval, assuming a Normal rather than t distribution (i.e. it is assumed that the required sample size will be large), is

Diagrammatically this can be represented as in Figure 9.1.

The firm wants the distance between X and ¡ to be no more than 20 pence in either direction, which means that the confidence interval must be 40 pence wide. The value of n which makes the confidence interval 40 pence wide has to be found. This can be done by solving the equation

20 = 2.58 x VsVn and hence by rearranging: