Question

A clothing store wants to determine the current percentage of customers who are over the age of forty. The store wants to create a 98% confidence interval which has a margin of error of at most 2 %. How many customers should be polled to create this confidence interval?

col1 Z0.10 Z0.05 Z0.025 Z0.01 Z0.005
col2 1.645 1.282 1.960 2.326 2.576

Answer 1

`n=(0.5(1-0.5))/(((0.02)/(2.326))^2)=3381.42`

And rounded up we have that n=3382

Explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

Solution to the problem

In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 98% of confidence, our significance level would be given by
`\alpha=1-0.98=0.02` and
`\alpha/2 =0.01`. And the critical value would be given by:

`z_(\alpha/2)=-2.326, t_(1-\alpha/2)=2.326`

The margin of error for the proportion interval is given by this formula:

`ME=z_(\alpha/2)\sqrt{(\hat p (1-\hat p))/(n)}` (a)

And on this case we have that
`ME =\pm 0.02` and we are interested in order to find the value of n, if we solve n from equation (a) we got:

`n=(\hat p (1-\hat p))/(((ME)/(z))^2)` (b)

Since we don't have prior information for the estimated proportion we can assume
`\hat p =0.5`. And replacing into equation (b) the values from part a we got:

`n=(0.5(1-0.5))/(((0.02)/(2.326))^2)=3381.42`

And rounded up we have that n=3382