Answer to Question #269185 in Statistics and Probability for syafiq

Question

The dean of a university wants to use the mean of a random sample to estimate the average amount of time students take to get from one class to the next. She wants to assert with probability 0.95 that her error will be at most 0.25 minutes. If she knows from the past studies that population standard deviation is 15 minutes, how large a sample she will need?

Answer 1

If $\bar{x}$ is used as an estimate of $\mu,$ we can be $100(1-\alpha)$% confident that the error $|\bar{x}-\mu|$

will not exceed a specified amount $E$ when the sample size is

$$n\geq(\dfrac{z_{\alpha/2}\sigma}{E})^2$$

Given 95% confidence interval, $z_{\alpha/2}=1.96$

$E=0.25\ min, \sigma=15\ min$

$$n\geq(\dfrac{1.96(15)}{0.25})^2$$

$$n\geq13830$$

$n=13830$