A researcher is interested in studying people’s mean age in a certain region. If the population standard deviation is known to be 8 years and 1.5 year of error margin is allowed, find the minimum simple size the researcher needs to use, knowing that he is going to conduct his study using 95% confidence level.
To determine the minimum sample size needed to estimate the population mean with a given level of confidence and margin of error, we can use the formula:
n = (Z * σ / E)^2
Where:
n = sample size
Z = z-score representing the desired confidence level
σ = population standard deviation
E = maximum allowable margin of error (half the width of the confidence interval)
In this case, the researcher wants to use a 95% confidence level, which means the Z-score will be the critical value for a confidence level of 95%. The critical value for a 95% confidence level is approximately 1.96.
Plugging in the given values:
σ = 8 (population standard deviation)
E = 1.5 (maximum allowable margin of error)
Z = 1.96 (critical value for a 95% confidence level)
n = (1.96 * 8 / 1.5)^2
n = (15.68 / 1.5)^2
n = 10.45^2
n ≈ 109.10
Therefore, the minimum sample size the researcher needs to use is approximately 110 (rounded up to the nearest whole number) to estimate the mean age with a 1.5-year margin of error at a 95% confidence level.