To calculate the 95% confidence interval for the true proportion of customers entering the bookstore because of the coffee shop, we’ll follow these steps:

1. Determine $\hat{p}$, the sample proportion.

Since you mentioned Sheet 1 of an Excel file, I assume $\hat{p}$ might be provided there, or we would need data from that sheet to calculate it. Without specific data, I'll explain the process:

$\hat{p} = \frac{\text{Number of customers who entered because of the coffee shop}}{\text{Total customers surveyed}}$

If, for example, 45 out of the 75 customers entered because of the coffee shop, then: $\hat{p} = \frac{45}{75} = 0.6$

2. Use the 95% confidence level critical value.

The critical $z^*$ value for a 95% confidence level, from your table, is $z^* = 1.960$.

3. Calculate the confidence interval.

The formula for a confidence interval for a proportion is:

$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

Where:

• $\hat{p}$ is the sample proportion,
• $z^*$ is the critical value (1.960),
• $n$ is the sample size (75).

For our example ($\hat{p} = 0.6$):

1. Calculate the margin of error (ME):

   $\text{ME} = 1.960 \times \sqrt{\frac{0.6 \times 0.4}{75}} = 1.960 \times \sqrt{0.0032} \approx 1.960 \times 0.0566 \approx 0.111$

2. Determine the interval:

   • Upper bound: $\hat{p} + \text{ME} = 0.6 + 0.111 = 0.711$
   • Lower bound: $\hat{p} - \text{ME} = 0.6 - 0.111 = 0.489$

Thus, the 95% confidence interval for the proportion is approximately $[0.489, 0.711]$.

Summary

The manager can say with 95% confidence that the true proportion of customers who come into the store because of the coffee shop is between 0.489 and 0.711.

Would you like further details on any step, or is there specific data in the Excel file that I should use?

Related Questions

1. How would the confidence interval change with a 90% or 99% confidence level?
2. What sample size would be required to narrow the interval to a margin of error of 0.05?
3. How does sample proportion variability affect the width of the confidence interval?
4. Can we use this confidence interval if the sample was not randomly selected?
5. What impact does increasing the sample size have on the confidence interval?

Tip

When constructing confidence intervals, remember that larger sample sizes reduce the margin of error, making the interval narrower and more precise.