You 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 5 years of the actual mean with a confidence level of 96%, how many citizens should be included in your sample? Assume that the standard deviation of the ages of all the citizens in this community is 20 years.
To determine the sample size needed for constructing a confidence interval, we can use the formula:
n = (Z * σ / E)^2
where:
n = sample size
Z = Z-score corresponding to the desired confidence level
σ = standard deviation of the population
E = margin of error (half of the desired interval width)
Here, the desired confidence level is 96% (which corresponds to a Z-score of 2.05 for a two-tailed test). The margin of error (E) is given as 5 years, and the standard deviation (σ) is 20 years.
Plugging these values into the formula:
n = (2.05 * 20 / 5)^2
n = (41 / 5)^2
n = 8.2^2
n ≈ 67.24
Rounding up to the nearest whole number, you should include at least 68 citizens in your sample.