Question

a magazine provided results from a poll of 2000 adults who were asked to identify their favorite pie. among the 2000 respondents, 14% chose chocolate pie, and the margin of error was given as ±4 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 sample proportion. the margin of error. the population proportion. found from evaluating \\( 1 - \hat { p } \\). the sample size. the value of p is

Explanation:

Step1: Determine \(\hat{p}\)

\(\hat{p}\) represents the sample proportion. Given that 14% chose chocolate pie, \(\hat{p}=0.14\)

Step2: Determine \(\hat{q}\)

\(\hat{q}\) is found from evaluating \(1 - \hat{p}\). So \(\hat{q}=1 - 0.14=0.86\)

Step3: Determine \(n\)

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

Step4: Determine \(E\)

\(E\) is the margin of error. Given as \(4\) percentage points, so \(E = 0.04\)

Step5: Determine \(p\)

\(p\) is the population proportion. Its value is unknown (we only have the sample proportion \(\hat{p}\))

Step6: Determine \(\alpha\)

The confidence level \(C=0.95\). Using the formula \(\alpha=1 - C\), we get \(\alpha=1 - 0.95 = 0.05\)

Answer:

\(\hat{p}=0.14\), \(\hat{q}=0.86\), \(n = 2000\), \(E = 0.04\), \(p\) (unknown population proportion), \(\alpha=0.05\)