a magazine provided results from a poll of 2000 adults who were asked to identify their favorite pie. among…

Explanation:

Step1: Define $\hat{p}$

$\hat{p}$ is the sample - proportion. Given that 12% of 2000 respondents chose chocolate pie, so $\hat{p}=0.12$.

Step2: Define $\hat{q}$

$\hat{q}=1 - \hat{p}$. Substituting $\hat{p}=0.12$, we get $\hat{q}=1 - 0.12 = 0.88$.

Step3: Define $n$

$n$ is the sample - size. Here, $n = 2000$ (the number of respondents in the poll).

Step4: Define $E$

$E$ is the margin of error. Given that the margin of error is $\pm3$ percentage points, so $E = 0.03$.

Step5: Define $p$

Since we have no prior information about the population - proportion $p$, we do not know its value. In the absence of other information, we can only estimate it using $\hat{p}$, but strictly speaking, its true value is unknown.

Step6: Calculate $\alpha$

The confidence level is 95%, so the confidence level $C=0.95$. We know that $\alpha=1 - C$. Substituting $C = 0.95$, we get $\alpha=1 - 0.95=0.05$.

Answer:

The value of $\hat{p}$ is $0.12$ The value of $\hat{q}$ is $0.88$ The value of $n$ is $2000$ The value of $E$ is $0.03$ The value of $p$ is unknown The value of $\alpha$ is $0.05$