Final answer:
To determine the sample size needed to construct a 99% confidence interval to estimate the population mean, use the formula n = (z^2 * σ^2) / EBM^2, where n is the sample size, z is the z-value, σ is the population standard deviation, and EBM is the margin of error.
Step-by-step explanation:
To determine the sample size needed to construct a 99% confidence interval to estimate the population mean, we can use the formula:
n = (z^2 * σ^2) / EBM^2
where n is the sample size, z is the z-value corresponding to the desired confidence level, σ is the population standard deviation, and EBM is the desired margin of error. In this case, σ = 79.
a) For a margin of error of 10:
n = (1.96^2 * 79^2) / 10^2 = 286.55
Rounding up to the next higher integer, the sample size needed is 287.
b) For a margin of error of 15:
n = (1.96^2 * 79^2) / 15^2 = 101.33
Rounding up to the next higher integer, the sample size needed is 102.
c) For a margin of error of 2:
n = (1.96^2 * 79^2) / 2^2 = 1203.52
Rounding up to the next higher integer, the sample size needed is 1204.