https://file.aiquickdraw.com/mathbot/file/17300797504868flak1gu.jpg

To calculate the 95% confidence interval for the proportion of customers who visit the store because of the coffee shop, we need the sample proportion (\(\hat{p}\)) and sample size. Here are the steps:

### Step-by-Step Solution

1. **Calculate Sample Proportion (\(\hat{p}\))**:
   - Use the data from the provided Excel file to determine the sample proportion, \(\hat{p}\), which represents the proportion of customers who came for the coffee shop.

2. **Determine the Confidence Level**:
   - For a 95% confidence level, the critical value \( z^* \) is given as \( 1.960 \).

3. **Formula for Confidence Interval**:
   - The confidence interval for a population proportion is calculated as:
     \[
     \hat{p} \pm z^* \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
     \]
     where:
     - \(\hat{p}\) is the sample proportion,
     - \( z^* = 1.960 \) (for 95% confidence),
     - \( n = 75 \) is the sample size.

4. **Compute the Margin of Error**:
   - Margin of error \( E = z^* \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \).

5. **Calculate Upper and Lower Bounds**:
   - The lower bound of the confidence interval is \(\hat{p} - E\),
   - The upper bound of the confidence interval is \(\hat{p} + E\).

Once you provide the sample proportion \(\hat{p}\) from the Excel file, I can help calculate the exact confidence interval.

Would you like further details on these steps, or do you have any questions?

---

Here are five related questions to deepen understanding:

1. How does changing the sample size affect the width of the confidence interval?
2. Why is the \( z \)-score 1.960 for a 95% confidence level?
3. How would the confidence interval change if the confidence level increased to 99%?
4. What is the effect of a larger sample proportion on the confidence interval?
5. Why is it important to know the margin of error in survey results?

**Tip:** In confidence interval calculations, a larger sample size generally results in a narrower interval, increasing the precision of the estimate.

The manager of a bookstore with a coffee shop wants to know the proportion of customers that come into the store because of the coffee shop. A random sample of 75 customers was polled. Use the given data to calculate the sample proportion (p̂) and the 95% confidence interval.

Learn how to calculate the 95% confidence interval for the proportion of customers visiting a bookstore because of its coffee shop. This problem involves key statistical concepts, including confidence intervals, margin of error, and the use of z-scores. Suitable for high school students or introductory college-level statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation