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:
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