Question

Mathematical Statistics with Applications Help Homework

Answer 1

7.43: Let $X$ denote the random variable for height and $\overline X$ for the sample mean. Then if $\mu$ is the mean of

So the probability that the difference between the sample and population means does not exceed 0.5 inch is

$P(|\overline X-\mu|\le0.5) = P\left(\left|\dfrac{\overline X-\mu}{0.25}\right|\le\dfrac{0.5}{0.25}\right) = P(|Z|\le 2) \approx \boxed{0.95}$

per the empirical or 68/95/99.7 rule.

7.44: For a sample of size n, the sample standard deviation would be $\frac{2.5}{\sqrt n}$. We want to find n such that

$P(|\overline X-\mu| < 0.4) = 0.95$

Comparing to the equation from the previous part, this means we would need

$\dfrac{0.4}{\frac{2.5}{\sqrt{n}}} = 0.16\sqrt n = 2 \implies \sqrt n = 12.5 \implies n = 156.25$

so a sample of at least 157 men would be sufficient.