Question

Use the given information to find the minimum sample size required to estimate an unknown population mean μ.

How many students must be randomly selected to estimate the mean weekly earnings of students at one college? We want 95% confidence that the sample mean is within $4 of the population mean, and the population standard deviation is known to be $25.

Answer 1

151 students must be selected.

Explanation:

We have that to find our
`\alpha` level, that is the subtraction of 1 by the confidence interval divided by 2. So:

`\alpha = (1 - 0.95)/(2) = 0.025`

Now, we have to find z in the Ztable as such z has a pvalue of
`1 - \alpha`.

That is z with a pvalue of
`1 - 0.025 = 0.975`, so Z = 1.96.

Now, find the margin of error M as such

`M = z(\sigma)/(√(n))`

In which
`\sigma` is the standard deviation of the population and n is the size of the sample.

The population standard deviation is known to be $25.

This means that
`\sigma = 25`

How many students must be randomly selected to estimate the mean weekly earnings of students at one college? Sample mean within $4 of the population mean

This is n for which M = 4. So

`M = z(\sigma)/(√(n))`

`4 = 1.96(25)/(√(n))`

`4√(n) = 1.96*25`

`√(n) = (1.96*25)/(4)`

`(√(n))^2 = ((1.96*25)/(4))^2`

`n = 150.1`

Rounding up:

151 students must be selected.