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Let's go through each of the variables in this question:

1. **Sample Proportion (\(\hat{p}\))**: This represents the proportion of respondents who chose chocolate pie. Since 11% of 1500 respondents chose chocolate pie, \(\hat{p} = 0.11\).

2. **Complement of Sample Proportion (\(\hat{q}\))**: This is calculated as \(1 - \hat{p}\), which represents the proportion of respondents who did not choose chocolate pie. So, \(\hat{q} = 1 - 0.11 = 0.89\).

3. **Sample Size (\(n\))**: This is the total number of respondents in the survey, which is given as \(n = 1500\).

4. **Margin of Error (E)**: The margin of error is given as \(\pm 5\%\), so \(E = 0.05\).

5. **Significance Level (\(\alpha\))**: The confidence level is 90%, which means the significance level \(\alpha\) is \(1 - 0.90 = 0.10\).

### Summary of Answers:

- The value of \(\hat{p}\) is **0.11**
- The value of \(\hat{q}\) is **0.89**
- The value of \(n\) is **1500**
- The value of \(E\) is **0.05**
- The value of \(\alpha\) is **0.10**

Would you like a deeper explanation on any part?

Here are some related questions for further practice:

1. How would the values change if the confidence level was 95%?
2. What is the formula for calculating the margin of error in a confidence interval?
3. How is the sample proportion different from the population proportion?
4. How does increasing the sample size affect the margin of error?
5. What confidence level would correspond to a significance level of 0.05?

**Tip:** When calculating confidence intervals, remember that the margin of error is based on both the sample proportion and the sample size.

A magazine provided results from a poll of 1500 adults who were asked to identify their favorite pie. Among the 1500 respondents, 11% chose chocolate pie, and the margin of error was given as ±5 percentage points. What values do p̂, q̂, n, E, and α represent? If the confidence level is 90%, what is the value of α?

Explore how to determine values of sample proportion (p̂), complement (q̂), sample size (n), margin of error (E), and significance level (α) in a poll of 1500 adults with a 90% confidence level. This problem introduces concepts in statistics, including calculating p̂ and q̂ from a sample, using margin of error, and deriving α from the confidence level.

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