Question

We intend to estimate the average driving time of a group of commuters. From a previous study, we believe that the average time is 42 minutes with a standard deviation of 10 minutes. We want our 90 percent confidence interval to have a margin of error of no more than plus or minus 2 minutes. What is the smallest sample size that we should consider

Answer 1

Answer:

The smallest sample size that we should consider is 68.

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:

$\alpha = \frac{1 - 0.9}{2} = 0.05$

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 $1 - 0.05 = 0.95$, so Z = 1.645.

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.

Standard deviation of 10 minutes.

This means that $\mu = 10$

What is the smallest sample size that we should consider?

This is n for which M = 2. So

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

$2 = 1.645\frac{10}{\sqrt{n}}$

$2\sqrt{n} = 1.645*10$

$\sqrt{n} = 1.645*5$

$(\sqrt{n})^2 = (1.645*5)^2$

$n = 67.7$

Rounding up:

The smallest sample size that we should consider is 68.