Question

In order to set​ rates, an insurance company is trying to estimate the number of sick days that full time workers at an auto repair shop take per year. A previous study indicated that the population standard deviation is 2.8 days. How large a sample must be selected if the company wants to be​ 95% confident that the true mean differs from the sample mean by no more than 1​ day?

A. 512
B. 1024
C. 141
D. 31

Answer 1

D. 31

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.

A previous study indicated that the population standard deviation is 2.8 days.

This means that
`\sigma = 2.8`

How large a sample must be selected if the company wants to be​ 95% 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 = 1.96(2.8)/(√(n))`

`√(n) = 1.96*2.8`

`(√(n))^2 = (1.96*2.8)^2`

`n = 30.12`

Rounding up, at least 31 people are needed, and the correct answer is given by option D.