Final answer:
To determine the required sample size for estimating the mean annual family medical expenses, we can use the formula for sample size calculation. Using a 90% confidence level and a margin of error of $60, the necessary sample size is approximately 117. If management wants to be correct to within $25, they would need to select a sample size of 608 employees.
Step-by-step explanation:
a. How large a sample is necessary?
To determine the sample size, we can use the formula for sample size calculation for estimating the mean:
n = (Z * σ / E)^2
Where n is the required sample size, Z is the standard score corresponding to the desired level of confidence (in this case, 90% confidence level corresponds to a Z-score of 1.645), σ is the standard deviation of the population (369), and E is the desired margin of error ($60).
Plugging in the values, we get:
n = (1.645 * 369 / 60)^2
n ≈ 116.32
Since the sample size must be a whole number, we round up to the nearest whole number. Therefore, a sample size of at least 117 is necessary.
b. If management wants to be correct to within (plus or minus $25), how many employees need to be selected?
Using the same formula as before, but with a margin of error of $25, we get:
n = (1.645 * 369 / 25)^2
n ≈ 607.61
Rounding up to the nearest whole number, management would need to select a sample size of 608 employees.