Question
Suppose a department store wants to estimate the mean age of the customers of its contemporary apparel department, correct to within 4 years, with level of confidence equal to 95%. Management believes that the standard deviation is 12 years. The sample size they should take is ________.
Answers
GPT-4o mini
To estimate the mean age of customers with a specified level of confidence and margin of error, we can use the formula for the required sample size \( n \):
\[
n = \left( \frac{Z \cdot \sigma}{E} \right)^2
\]
Where:
- \( Z \) = Z-value corresponding to the desired confidence level
- \( \sigma \) = population standard deviation
- \( E \) = margin of error (the amount we want to be within the true mean)
For a 95% confidence level, the Z-value is approximately 1.96.
Given:
- Standard deviation (\( \sigma \)) = 12 years
- Margin of error (\( E \)) = 4 years
Substituting these values into the formula:
\[
n = \left( \frac{1.96 \cdot 12}{4} \right)^2
\]
Calculating the numerator:
\[
1.96 \cdot 12 = 23.52
\]
Now, divide by the margin of error (\( E \)):
\[
\frac{23.52}{4} = 5.88
\]
Now square that:
\[
n = (5.88)^2 = 34.5744
\]
Since you cannot have a fraction of a sample, we round up to the next whole number:
\[
n = 35
\]
Therefore, the department store should take a sample size of **35** customers to estimate the mean age of the customers in its contemporary apparel department, correct to within 4 years, with a 95% confidence level.
\[
n = \left( \frac{Z \cdot \sigma}{E} \right)^2
\]
Where:
- \( Z \) = Z-value corresponding to the desired confidence level
- \( \sigma \) = population standard deviation
- \( E \) = margin of error (the amount we want to be within the true mean)
For a 95% confidence level, the Z-value is approximately 1.96.
Given:
- Standard deviation (\( \sigma \)) = 12 years
- Margin of error (\( E \)) = 4 years
Substituting these values into the formula:
\[
n = \left( \frac{1.96 \cdot 12}{4} \right)^2
\]
Calculating the numerator:
\[
1.96 \cdot 12 = 23.52
\]
Now, divide by the margin of error (\( E \)):
\[
\frac{23.52}{4} = 5.88
\]
Now square that:
\[
n = (5.88)^2 = 34.5744
\]
Since you cannot have a fraction of a sample, we round up to the next whole number:
\[
n = 35
\]
Therefore, the department store should take a sample size of **35** customers to estimate the mean age of the customers in its contemporary apparel department, correct to within 4 years, with a 95% confidence level.