Question

Suppose we want to determine the minimum sample size required to give us a 95% confidence interval that estimates, to within $500, the average salary of a state employee. Also, suppose that from a previous experiment, we know that the population standard deviation is $6300. What is the minimum sample size required

Answer 1

The minimum sample size is 610.

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`.

So it is z with a pvalue of
`1-0.025 = 0.975`, so
`z = 1.96`

Now, find 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.

Population standard deviation is $6300, so
`\sigma = 6300`

Suppose we want to determine the minimum sample size required to give us a 95% confidence interval that estimates, to within $500, the average salary of a state employee.

This is n when M = 500. So

`M = z*(\sigma)/(√(n))`

`500 = 1.96*(6300)/(√(n))`

`500√(n) = 1.96*6300`

`√(n) = (1.96*6300)/(500)`

`(√(n))^(2) = ((1.96*6300)/(500))^(2)`

`n = 609.89`

Rouding up

The minimum sample size is 610.