To calculate the required sample size for estimating the mean age of the customers, we can use the formula for the sample size in estimating a population mean:
\[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \]
Where:
- \( n \) = sample size
- \( Z \) = Z-value corresponding to the desired confidence level
- \( \sigma \) = standard deviation of the population
- \( E \) = margin of error (the desired accuracy)
Given:
- Confidence level = 95% → Z-value for 95% confidence is approximately 1.96
- Standard deviation (\( \sigma \)) = 8 years
- Margin of error (\( E \)) = 2 years
Now we plug in the values into the formula:
\[ n = \left( \frac{1.96 \cdot 8}{2} \right)^2 \]
Calculating the numerator:
\[ 1.96 \cdot 8 = 15.68 \]
Now dividing by the margin of error:
\[ \frac{15.68}{2} = 7.84 \]
Now squaring that result:
\[ n = (7.84)^2 = 61.4656 \]
Since sample size must be a whole number, we round up to the next whole number:
\[ n \approx 62 \]
Therefore, the department store should take a sample size of 62 customers to estimate the mean age of the customers of its contemporary apparel department, correct to within 2 years at a 95% confidence level.