Questions: A magazine provided results from a poll of 1500 adults who were asked to identify their favorite pie. Among the 1500 respondents, 12% chose chocolate pie, and the margin of error was given as ± 5 percentage points. What values do p̂, q̂, n, E, and p represent? If the confidence level is 95%, what is the value of α? The value of p̂ is the sample proportion. The value of q̂ is found from evaluating 1-p̂. The value of n is the sample size. The value of E is the margin of error. The value of p is the population proportion.

A magazine provided results from a poll of 1500 adults who were asked to identify their favorite pie. Among the 1500 respondents, 12% chose chocolate pie, and the margin of error was given as ± 5 percentage points. What values do p̂, q̂, n, E, and p represent? If the confidence level is 95%, what is the value of α?

The value of p̂ is the sample proportion.
The value of q̂ is found from evaluating 1-p̂.
The value of n is the sample size.
The value of E is the margin of error.
The value of p is the population proportion.

Transcript text: A magazine provided results from a poll of 1500 adults who were asked to identify their favorite pie. Among the 1500 respondents, $12 \%$ chose chocolate pie, and the margin of error was given as $\pm 5$ percentage points. What values do $\hat{p}, \hat{q}, n, E$, and $p$ represent? If the confidence level is $95 \%$, what is the value of $\alpha$ ? The value of $\hat{p}$ is The value of $\hat{q}$ is The value of $n$ is The value of $E$ is The value of $p$ is found from evaluating $1-\hat{\mathbf{p}}$. the sample proportion. the sample size. the population proportion. the margin of error.

Solution

Solution Steps

Step 1: Calculate the Sample Proportion ($\hat{p}$)

To calculate the sample proportion $\hat{p}$, divide the number of respondents with the characteristic by the total sample size $n$. $$\hat{p} = \frac{180}{1500} = 0.12$$

Step 2: Determine the Complement of the Sample Proportion ($\hat{q}$)

The complement of the sample proportion $\hat{q}$ is calculated as $1 - \hat{p}$. $$\hat{q} = 1 - \hat{p} = 1 - 0.12 = 0.88$$

Step 3: Identify the Sample Size ($n$)

The sample size $n$ is directly identified from the information provided and is 1500.

Step 4: Note the Margin of Error ($E$)

The margin of error $E$ is given in the problem statement as 0.05.

Step 5: The Population Proportion ($p$)

The population proportion $p$ is typically what one aims to estimate and is not directly calculated but inferred from $\hat{p}$ and $E$.

Step 6: Calculate the Significance Level ($\alpha$)

To find the significance level $\alpha$, subtract the confidence level (expressed as a decimal) from 1. $$\alpha = 1 - \text{Confidence Level} = 1 - 0.95 = 0.05$$

Final Answer:

The sample proportion $\hat{p}$ is 0.12, the complement $\hat{q}$ is 0.88, the sample size $n$ is 1500, the margin of error $E$ is 0.05, and the significance level $\alpha$ is 0.05.

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