To estimate the mean age of customers within a specified margin of error with a given level of confidence, we can use the formula for the sample size (\( n \)) required for estimating a population mean:
\[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \]
Where:
- \( Z \) is the z-value corresponding to the desired confidence level,
- \( \sigma \) is the population standard deviation,
- \( E \) is the margin of error.
Given:
- Level of confidence = 95%, which corresponds to a z-value (\( Z \)) of approximately 1.96 (since the critical value for a 95% confidence level is 1.96).
- Standard deviation (\( \sigma \)) = 8 years.
- Margin of error (\( E \)) = 2 years.
Now, substituting the known values into the formula:
\[ n = \left( \frac{1.96 \cdot 8}{2} \right)^2 \]
Calculating the inside of the parentheses first:
\[ \frac{1.96 \cdot 8}{2} = \frac{15.68}{2} = 7.84 \]
Now squaring that result:
\[ n = (7.84)^2 = 61.4656 \]
Since the sample size must be a whole number, we round up to the nearest whole number:
\[ n \approx 62 \]
Therefore, the department store should take a sample size of 62 customers.