Question

A corporation with several thousand employees wants to estimate the mean commute time for all employees. They would like to construct a 95% confidence interval with a margin of error of no more than 4 minutes. Preliminary interviews with a small sample suggest that a reasonable estimate of the population standard deviation is σ = 10 minutes. Which of the following is the smallest sample the company can take to achieve the desired margin or error?

A.5
B.24
C.25
D.41
E.42

Answer 1

C.25

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 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.

Which of the following is the smallest sample the company can take to achieve the desired margin or error?

This is n when
`\sigma = 10, M = 4`

`M = z*(\sigma)/(√(n))`

`4 = 1.96*(10)/(√(n))`

`4√(n) = 1.96*10`

`√(n) = (1.96*10)/(4)`

`(√(n))^(2) = ((1.96*10)/(4))^(2)`

`n = 24.01`

We have to round up.

So the correct answer is:

C.25