Question

a magazine provided results from a poll of 2000 adults who were asked to identify their favorite pie. among the 2000 respondents, 11% chose chocolate pie, and the margin of error was given as ±5 percentage points. what values do \\( \hat { p }, \hat { q }, n, e \\), and \\( p \\) represent? if the confidence level is 99%, what is the value of \\( \alpha \\)?

Explanation:

Step1: Determine the value of \(\hat{p}\)

\(\hat{p}\) is the sample proportion. Given that 11% chose chocolate pie, so \(\hat{p}=0.11\)

Step2: Determine the value of \(\hat{q}\)

\(\hat{q}\) is found from evaluating \(1 - \hat{p}\). So \(\hat{q}=1 - 0.11=0.89\)

Step3: Determine the value of \(n\)

\(n\) is the sample size. Here, \(n = 2000\)

Step4: Determine the value of \(E\)

\(E\) is the margin of error. Given \(E = 0.05\) (since \(5\) percentage points \(=0.05\))

Step5: Determine the value of \(\alpha\)

The confidence level \(C=0.99\). Using the formula \(\alpha=1 - C\), we have \(\alpha=1 - 0.99 = 0.01\)

Step6: About \(p\)

\(p\) is the population proportion. Its value is unknown (we are trying to estimate it using the sample proportion \(\hat{p}\))

Answer:

The value of \(\hat{p}\) is \(0.11\) (the sample proportion).
The value of \(\hat{q}\) is \(0.89\) (found from evaluating \(1-\hat{p}\)).
The value of \(n\) is \(2000\) (the sample size).
The value of \(E\) is \(0.05\) (the margin of error).
The value of \(p\) is unknown (the population proportion).
The value of \(\alpha\) is \(0.01\)