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

The smallest sample size that we should consider is 68.

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 - 0.9)/(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(\sigma)/(√(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(\sigma)/(√(n))`

`2 = 1.645(10)/(√(n))`

`2√(n) = 1.645*10`

`√(n) = 1.645*5`

`(√(n))^2 = (1.645*5)^2`

`n = 67.7`

Rounding up:

The smallest sample size that we should consider is 68.