To solve this problem, we need to calculate the 95% confidence interval for the estimated proportion (denoted [tex]\(\hat{p}\)[/tex]) of customers that come into the store because of the coffee shop.
Here is the step-by-step solution:
1. Identify the Sample Size and Sample Proportion:
- The sample size ([tex]\(n\)[/tex]) is 75.
- The sample proportion ([tex]\(\hat{p}\)[/tex]) is 0.123.
2. Determine the Z-Score for the Desired Confidence Level:
- For a 95% confidence interval, the Z-score ([tex]\(z^*\)[/tex]) is 1.960. This value is obtained from the standard normal distribution corresponding to a 95% confidence level.
3. Calculate the Standard Error:
- The standard error (SE) of the sample proportion is calculated using the formula:
[tex]\[
SE = \sqrt{\frac{\hat{p} \cdot (1 - \hat{p})}{n}}
\][/tex]
- Plugging in the values:
[tex]\[
SE = \sqrt{\frac{0.123 \cdot (1 - 0.123)}{75}}
\][/tex]
4. Calculate the Margin of Error:
- The margin of error (ME) is given by:
[tex]\[
ME = z^* \cdot SE
\][/tex]
- Using the values:
[tex]\[
ME = 1.960 \cdot SE
\][/tex]
5. Determine the Confidence Interval:
- The lower bound of the confidence interval is:
[tex]\[
\text{Lower bound} = \hat{p} - ME
\][/tex]
- The upper bound of the confidence interval is:
[tex]\[
\text{Upper bound} = \hat{p} + ME
\][/tex]
6. Calculate the Numerical Values:
- After calculating, we find that the standard error is approximately [tex]\(0.038\)[/tex].
- The margin of error then is approximately [tex]\(0.074\)[/tex].
- Therefore, the confidence interval is:
[tex]\[
\text{Lower bound} = 0.123 - 0.074 = 0.049
\][/tex]
[tex]\[
\text{Upper bound} = 0.123 + 0.074 = 0.197
\][/tex]
Thus, the manager can say with 95% confidence that the true population proportion of customers that come into the store because of the coffee shop is in the interval [tex]\((0.049, 0.197)\)[/tex].
### Summary
The 95% confidence interval for the proportion of customers that come into the store because of the coffee shop is:
- Upper bound: 0.197
- Lower bound: 0.049
Therefore, you can state that with 95% confidence, the true population proportion is in the interval (0.049, 0.197).
