A magazine provided results from a poll of 1500 adults who were asked to identify their favorite pie. Among the 1500 ​respondents, 13​% chose chocolate​ pie, and the margin of error was given as plus or minus5 percentage points. What values do ModifyingAbove p with caret​, ModifyingAbove q with caret​, ​n, E, and p​ represent? If the confidence level is 95​%, what is the value of alpha​?

In this problem, you're provided with results from a poll and are asked to determine various values related to the sample proportion and confidence level. Let's define and calculate these values step by step.

1. **ModifyingAbove p with caret ( \(\hat{p}\)**): This represents the **sample proportion** of respondents who chose chocolate pie.
   - \(\hat{p} = \frac{13\%}{100} = 0.13\)

2. **ModifyingAbove q with caret ( \(\hat{q}\)**): This represents the proportion of respondents who did **not** choose chocolate pie. It's simply the complement of \(\hat{p}\), so:
   - \(\hat{q} = 1 - \hat{p} = 1 - 0.13 = 0.87\)

3. **n**: This represents the **sample size**, which is given as 1500.
   - \(n = 1500\)

4. **E (Margin of Error)**: This is the maximum expected difference between the sample proportion and the true population proportion. It is given as ±5 percentage points.
   - \(E = 5\%\) or 0.05

5. **p (True Population Proportion)**: The true population proportion \(p\) is **unknown** in this context. The sample proportion \(\hat{p}\) is an estimate of \(p\), but we do not know the exact value of \(p\).

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Now, let's determine the **value of alpha ( \(\alpha\)**):

- The **confidence level** is given as 95%. The confidence level and alpha (\(\alpha\)) are related as follows:
   - \(\alpha = 1 - \text{Confidence Level}\)
   - \(\alpha = 1 - 0.95 = 0.05\)

Therefore, the value of \(\alpha\) is **0.05**.

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### Summary:
- \(\hat{p} = 0.13\)
- \(\hat{q} = 0.87\)
- \(n = 1500\)
- \(E = 0.05\)
- \(\alpha = 0.05\)

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Do you want any further details or have questions on these calculations?

Here are five related questions to expand your understanding:
1. How do we calculate the confidence interval based on the margin of error?
2. How can we interpret the meaning of a 95% confidence level?
3. What effect does increasing the sample size have on the margin of error?
4. How is the margin of error derived using standard error and the z-score?
5. What does the alpha value tell us about the probability of making a Type I error?

**Tip**: In most cases, increasing the sample size will reduce the margin of error, leading to a more precise estimate of the population parameter.

A magazine provided results from a poll of 1500 adults who were asked to identify their favorite pie. Among the 1500 respondents, 13% chose chocolate pie, and the margin of error was given as plus or minus 5 percentage points. What values do ModifyingAbove p with caret​, ModifyingAbove q with caret​, ​n, E, and p​ represent? If the confidence level is 95​%, what is the value of alpha​?

Learn how to analyze a poll of 1500 adults identifying their favorite pie with 13% choosing chocolate pie. This problem covers key statistical concepts like sample proportions, margin of error, and alpha calculation at a 95% confidence level. Ideal for students in Grades 10-12.

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