8.4 Calculating the sample size n: continuous random variables and

Continuous random variables

If we go back to our standardizing formula for the sampling distribution for means, we can see that it is possible to solve it for n. If we do this we have $(\bar{X}-\mu )$ in the denominator.

Because we have not taken a sample yet we do not know any of the variables in the formula except that we can set Z _α to the level of confidence we desire just as we did when determining confidence intervals. If we set a predetermined acceptable error, or tolerance, for the difference between $\bar{X}$ and μ, called e in the formula, we are much further in solving for the sample size n. We still do not know the population standard deviation, σ. In practice, a pre-survey is usually done which allows for fine tuning the questionnaire and will give a sample standard deviation that can be used. In other cases, previous information from other surveys may be used for σ in the formula. While crude, this method of determining the sample size may help in reducing cost significantly. It will be the actual data gathered that determines the inferences about the population, so caution in the sample size is appropriate calling for high levels of confidence and small sampling errors.

Binary random variables

where e = (p′-p), and is the acceptable sampling error, or tolerance, for the application.

In this case the very object of our search is in the formula, p, and of course q because q =1-p. This result occurs because the binomial distribution is a one parameter distribution. If we know p then we know the mean and the standard deviation. Therefore, p shows up in the standard deviation of the sampling distribution which is where we got this formula. If, in an abundance of caution, we substitute 0.5 for p we will draw the largest required sample size that will provide the level of confidence specified by Zα. This is true because of all combinations of two numbers that add to one, the largest multiple is when each is 0.5. Without any other information concerning the population parameter p, this is the common practice. This may result in oversampling, but certainly not under sampling, thus, this is a cautious approach.

There is an interesting trade-off between the level of confidence and the sample size that shows up here when considering the cost of sampling. [link] shows the appropriate sample size at different levels of confidence and different level of the acceptable error, or tolerance.