Suppose that the anthropologist of Exercise 7.43 wants the difference between the sample mean and the population mean to be less than 0.4 inch, with probability 0.95. How many men should she sample to achieve this objective?

Exercise 7.43
An anthropologist wishes to estimate the average height of men for a certain race of people. If the population standard deviation is assumed to be 2.5 inches and if she randomly samples 100 men, find the probability that the difference between the sample mean and the true population mean will not exceed 0.5 inch.


Given that the sample standard deviation $s =2.5$, margin of error $E =0.4$. The confidence coefficient is $1-\alpha=0.95$. Thus $\alpha = 0.05$.

Z-critical value
Z-critical value

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

The minimum sample size required to estimate the mean is

$$ \begin{aligned} n &= \bigg(\frac{z* s}{E}\bigg)^2\\ & = \bigg(\frac{1.96*2.5}{0.4}\bigg)^2\\ & =150.0625\\ &\approx 151. \end{aligned} $$

Thus, the sample of size $n=151$ will ensure that the $95$% confidence interval for the mean will have a margin of error $0.4$.

Further Reading