Question

ou are interested in estimating the the mean age of the citizens living in your community. In order to do this, you plan on constructing a confidence interval; however, you are not sure how many citizens should be included in the sample. If you want your sample estimate to be within 4 years of the actual mean with a confidence level of 96%, how many citizens should be included in your sample

Answer 1

`((4\sigma)/(2.056))^2`, rounded up, if needed, citizens should be included in the sample, in which
`\sigma` is the standard deviation of the population.

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 - 0.96)/(2) = 0.02`

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 - 0.02 = 0.98`, so Z = 2.056.

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.

Within 4 years of the actual mean

We have to find n for which M = 4. So

`M = z(\sigma)/(√(n))`

`4 = 2.056(\sigma)/(√(n))`

`2.056√(n) = 4\sigma`

`√(n) = (4\sigma)/(2.056)`

`(√(n))^2 = ((4\sigma)/(2.056))^2`

`n = ((4\sigma)/(2.056))^2`

`((4\sigma)/(2.056))^2`, rounded up, if needed, citizens should be included in the sample, in which
`\sigma` is the standard deviation of the population.