Question

4)In order to set rates, an insurance company is trying to estimate the number of sick daysthat full time workers at an auto repair shop take per year. A previous study indicated thatthe standard deviation was2.2 days. a) How large a sample must be selected if thecompany wants to be 92% confident that the true mean differs from the sample mean by nomore than 1 day

Answer 1

A sample of 18 is required.

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 - 0.92)/(2) = 0.04`

Now, we have to find z in the Z-table as such z has a p-value of
`1 - \alpha`.

That is z with a pvalue of
`1 - 0.04 = 0.96`, so Z = 1.88.

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.

A previous study indicated that the standard deviation was 2.2 days.

This means that
`\sigma = 2.2`

How large a sample must be selected if the company wants to be 92% confident that the true mean differs from the sample mean by no more than 1 day?

This is n for which M = 1. So

`M = z(\sigma)/(√(n))`

`1 = 1.88(2.2)/(√(n))`

`√(n) = 1.88*2.2`

`(√(n))^2 = (1.88*2.2)^2`

`n = 17.1`

Rounding up:

A sample of 18 is required.