Let’s walk through the steps to determine the 95% confidence interval for the proportion of customers who come into the store because of the coffee shop, using the given data.
1. Determine the Point Estimate ([tex]\(\hat{p}\)[/tex]):
- The sample proportion [tex]\(\hat{p}\)[/tex] given is 0.743.
2. Sample Size (n):
- The sample size [tex]\(n\)[/tex] is 75.
3. Confidence Level:
- We are working with a 95% confidence interval. From the provided table, we know that the Z-value ([tex]\(z^*\)[/tex]) for a 95% confidence interval is 1.960.
4. Calculate the Standard Error (SE):
- The formula for the standard error of the sample proportion is:
[tex]\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\][/tex]
- Substituting the given values:
[tex]\[
SE = \sqrt{\frac{0.743 \times (1 - 0.743)}{75}} \approx 0.050
\][/tex]
5. Calculate the Margin of Error (ME):
- The margin of error is calculated using the formula:
[tex]\[
ME = z^* \times SE
\][/tex]
- Substituting the values:
[tex]\[
ME = 1.960 \times 0.050 \approx 0.099
\][/tex]
6. Determine the Confidence Interval:
- The confidence interval is found by adding and subtracting the margin of error from the sample proportion.
- The lower bound of the confidence interval is:
[tex]\[
\hat{p} - ME = 0.743 - 0.099 = 0.644
\][/tex]
- The upper bound of the confidence interval is:
[tex]\[
\hat{p} + ME = 0.743 + 0.099 = 0.842
\][/tex]
7. Conclusion:
- The 95% confidence interval for the proportion of customers that come into the store because of the coffee shop is (0.644, 0.842), rounded to three decimal places.
Thus, the final answers are:
- Lower bound for 95% confidence interval = 0.644
- Upper bound for 95% confidence interval = 0.842
