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.