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, $z_{\alpha/2}=1.96$

$E=0.25\ min, \sigma=1.5\ min$

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

$$n\geq13830$$

$n=13830$