what is the probability that the contractor's estimate will be within 5 weeks of the true mean

Question

A contractor uses sample mean lifetime $x'$ of $250$ compressors as her estimate for population mean lifetime m of all new compressors. If this brand of compressors has a standard deviation of $35$ months, what is the probability that the contractor's estimate will be within $5$ weeks of the true mean

Answer 1

How many weeks are there in a month? The answer will depend on that! We will use $4$, despite the fact that it is almost always wrong. So we take $5$ weeks to be $\frac{5}{4}$ months.

Let $\bar{X}$ be the sample mean. Since $\bar{X}$ is a sum of a large number of identically distributed independent random variables, it is reasonable to suppose that $\bar{X}$ has a close to normal distribution.

The standard deviation of $\bar{X}$ is $\frac{35}{\sqrt{250}}$. For brevity, call this $c$. Let $\mu$ be the true mean. Then $$\Pr\left(|\bar{X}-\mu|\lt \frac{5}{4}\right)=\Pr\left(\left|\frac{\bar{X}-\mu}{c}\right|\lt \frac{5}{4c}\right).\tag{1}$$

The probability on the right of (1) is approximately $\Pr(|Z|\lt \frac{5}{4c})$, where $Z$ is standard normal. Now we can use tables, or software, to finish.