Suppose that a researcher is interested in estimating the mean systolic blood pressure, $\mu$, of executives of major corporations. He plans to use the blood pressures of a random sample of executives of major corporations to estimate m. Assuming that the standard deviation of the population of systolic blood pressures of executives of major corporations is 24 mm Hg, what is the minimum sample size needed for the researcher to be 95% confident that his estimate is within 6 mm Hg of mean $\mu$?

#### Solution

Given that the margin of error $E =6$. The confidence coefficient is $1-\alpha=0.95$. Thus $\alpha = 0.05$.

The population standard deviation is $\sigma = 24$.

The critical value of $Z$ is $z=Z_{\alpha/2} = 1.96$.

The minimum sample size required to estimate the mean is

` $$ \begin{aligned} n &= \bigg(\frac{z* \sigma}{E}\bigg)^2\\ & = \bigg(\frac{1.96*24}{6}\bigg)^2\\ & =61.4656\\ &\approx 62. \end{aligned} $$ `

Thus, the sample of size $n=62$ will ensure that the $95$% confidence interval for the mean will have a margin of error $6$.