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 mean, find the probability that the difference between the sample mean and the true population mean will not exceed 0.5 inch.
[Math] Find the probability that the difference between the sample mean and the true population mean wll not exceed 0.5 inch
statistics
Best Answer
We will assume that by "difference does not exceed $0.5$ inch," what is meant is that the absolute value of the difference does not exceed $0.5$ inch.
Let the heights of the people in the sample be $X_1,X_2,\dots,X_{100}$. Then the sample mean is the random variable $Y$, where $$Y=\frac{1}{100}(X_1+X_2+\cdots+X_{100}).$$ Let the population mean be $mu$. Then $Y$ has mean $\mu$ and standard deviation $\frac{2.5}{\sqrt{100}}$.
We assume that the distribution of the $X_i$ is reasonably nice. Then $Y$ has a close to normal distribution. From here on, we assume the distribution is normal. Then $Y-\mu$ is normal with mean $0$ and standard deviation $0.25$. Then $$\Pr\left(|Y-\mu|\le 0.5\right)=\Pr\left(|Z|\le \frac{0.5}{0.25}\right).$$ This can be found using tables of the standard normal, or software.