To find the 95% confidence interval for the proportion of customers who visit the bookstore because of the coffee shop, follow these steps:
1. Identify the Sample Proportion [tex]\(\hat{p}\)[/tex]:
The sample proportion, [tex]\(\hat{p}\)[/tex], is given as 0.123 from the random sample of 75 customers.
2. Determine the Sample Size [tex]\(n\)[/tex]:
The sample size is given as [tex]\(n = 75\)[/tex].
3. Identify the Z-Score for the Desired Confidence Level:
For a 95% confidence level, the corresponding Z-score ([tex]\(z^*\)[/tex]) is 1.960.
4. Calculate the Standard Error (SE):
The standard error is calculated using the formula:
[tex]\[
SE = \sqrt{\frac{\hat{p} (1-\hat{p})}{n}}
\][/tex]
Plugging in the given values:
[tex]\[
SE = \sqrt{\frac{0.123 \cdot (1 - 0.123)}{75}} = 0.038
\][/tex]
5. Calculate the Margin of Error (MoE):
The margin of error is found by multiplying the Z-score by the standard error:
[tex]\[
MoE = z^* \cdot SE = 1.960 \cdot 0.038 = 0.074
\][/tex]
6. Calculate the Lower and Upper Bounds of the Confidence Interval:
- Lower Bound:
[tex]\[
\text{Lower Bound} = \hat{p} - MoE = 0.123 - 0.074 = 0.049
\][/tex]
- Upper Bound:
[tex]\[
\text{Upper Bound} = \hat{p} + MoE = 0.123 + 0.074 = 0.197
\][/tex]
7. Conclusion:
The manager can say with 95% confidence that the true population proportion of customers who come into the store because of the coffee shop is in the interval:
[tex]\[
(0.049, 0.197)
\][/tex]
Rounded to three decimal places, the lower bound is 0.049 and the upper bound is 0.197.
