8.2.2.3  Computing Necessary Sample Size
8.2.2.3  Computing Necessary Sample SizeCalculating the sample size necessary for estimating a population mean with a given margin of error and level of confidence is similar to that for estimating a population proportion. However, since the \(t\) distribution is not as “neat” as the standard normal distribution, the process can be iterative. (Recall, the shape of the \(t\) distribution is different for each degree of freedom). This means that we would solve, reset, solve, reset, etc. until we reached a conclusion. Yet, we can avoid this iterative process if we employ an approximate method based on \(t\) distribution approaching the standard normal distribution as the sample size increases. This approximate method invokes the following formula:
 Finding the Sample Size for Estimating a Population Mean
 \(n=\dfrac{z^{2}\widetilde{\sigma}^{2}}{M^{2}}=\left ( \dfrac{z\widetilde{\sigma}}{M} \right )^2\)

\(z\) = z multiplier for given confidence level
\(\widetilde{\sigma}\) = estimated population standard deviation
\(M\) = margin of error
The sample standard deviation may be estimated on the basis of prior research studies.
8.2.2.3.1  Example: Estimating IQ
8.2.2.3.1  Example: Estimating IQExample: Estimating IQ
A team of researchers wants to estimate the mean IQ of students enrolled at one prestigious university. Previous research studies have examined samples of students from other similar universities and usually find results around \(\overline{x}=120\) and \(s=10\). In order to construct a 90% confidence interval with a margin of error of \(\pm2\) IQ points, what sample size should be obtained?
As shown in the probability distribution plot below, the z value associated with a 90% confidence interval is 1.645.
The estimated standard deviation is given to be 10 and the desired margin of error is given to be 2.
\(n=\dfrac{z^{2}\widetilde{\sigma}^{2}}{M^{2}}=\dfrac{1.645^{2}(10^{2})}{2^{2}}=67.615\)
We round up to 68. The research team should attempt to obtain a sample of at least 68 individuals.