The dwarves of the Grey Mountains wish to conduct a survey of their pick-axes in order to construct a 95% confidence interval about the proportion of pick-axes in need of repair. What minimum sample size would be necessary in order ensure a margin of error of 2 percentage points (or less) if they do not have a prior estimate of the percentage of pick-axes that are in need of repair?

The minimum sample size needed is 2401.

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of , and a confidence level of , we have the following confidence interval of proportions.

In which

z is the zscore that has a pvalue of .

The margin of error is:

95% confidence level

So , z is the value of Z that has a pvalue of , so .

What minimum sample size would be necessary in order ensure a margin of error of 2 percentage points (or less) if they do not have a prior estimate of the percentage of pick-axes that are in need of repair?

The minimum sample size needed is n.

n is found when

We do not have a prior estimate of the percentage of pick-axes that are in need of repair, so we use

The minimum sample size needed is 2401.