Step 1 :In this problem, we are given a poll of 1000 adults where 12% chose chocolate pie. The margin of error is given as ±3 percentage points. We are asked to identify the values of \(\hat{p}\), \(\hat{q}\), \(n\), \(E\), and \(p\).
Step 2 :\(\hat{p}\) represents the sample proportion, which in this case is the proportion of adults who chose chocolate pie. So, \(\hat{p}\) = 0.12
Step 3 :\(\hat{q}\) represents the complement of the sample proportion, which is the proportion of adults who did not choose chocolate pie. So, \(\hat{q}\) = 1 - \(\hat{p}\) = 1 - 0.12 = 0.88
Step 4 :\(n\) represents the sample size, which is the total number of adults polled. So, \(n\) = 1000
Step 5 :\(E\) represents the margin of error, which is the range within which the true population proportion is likely to fall. So, \(E\) = 0.03
Step 6 :\(p\) represents the population proportion, which is the true proportion of all adults who would choose chocolate pie. However, this value is unknown in this problem.
Step 7 :The confidence level is 95%, so \(\alpha\) would be 1 - 0.95 = 0.05, representing the probability that the true population proportion falls outside the confidence interval.
Step 8 :Final Answer: The value of \(\hat{p}\) is \(\boxed{0.12}\), The value of \(\hat{q}\) is \(\boxed{0.88}\), The value of \(n\) is \(\boxed{1000}\), The value of \(E\) is \(\boxed{0.03}\), The value of \(p\) is unknown, The value of \(\alpha\) is \(\boxed{0.05}\)