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

Answer:

C.25

Step-by-step explanation:

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

$\alpha = \frac{1-0.95}{2} = 0.025$

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

$M = z*\frac{\sigma}{\sqrt{n}}$

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 $\sigma = 10, M = 4$

$M = z*\frac{\sigma}{\sqrt{n}}$

$4 = 1.96*\frac{10}{\sqrt{n}}$

$4\sqrt{n} = 1.96*10$

$\sqrt{n} = \frac{1.96*10}{4}$

$(\sqrt{n})^{2} = (\frac{1.96*10}{4})^{2}$

$n = 24.01$

We have to round up.

So the correct answer is:

C.25