Number skills of young men. Suppose that scores of men aged 21 to 25 years on the quantitative part of the National Assessment of Education Progress (NAEP) test follow a Normal distribution with standard deviation $\sigma= 60$. You want to estimate the mean score within $\pm$ 10 with 90% confidence. How large an SRS of scores must you choose?


Given that the sample standard deviation $s =60$, margin of error $E =10$. The confidence coefficient is $1-\alpha=0.9$. Thus $\alpha = 0.1$.

Z-critical value
Z-critical value

The critical value of $Z$ is $z=Z_{\alpha/2} = 1.64$.

The minimum sample size required to estimate the mean is

$$ \begin{aligned} n &= \bigg(\frac{z* s}{E}\bigg)^2\\ & = \bigg(\frac{1.64*60}{10}\bigg)^2\\ & =96.8256\\ &\approx 97. \end{aligned} $$
Thus, the sample of size $n=97$ will ensure that the $90$% confidence interval for the mean will have a margin of error $10$.

Further Reading