Q.11 - The Practice of Statistics for AP

Question

Prayer in school A New York Times/CBS News Poll asked the question, “Do you favor an amendment to the Constitution that would permit organized prayer in public schools?” Sixty-six percent of the sample answered “Yes.” The article describing the poll says that it “is based on telephone interviews conducted from Sept. 13 to Sept. 18 with $1664$ adults around the United States, excluding Alaska and Hawaii. . . . The telephone numbers were formed by random digits, thus permitting access to both listed and unlisted residential numbers.” The article gives the margin of error for a $95\%$ confidence level as $3$ percentage points.

(a) Explain what the margin of error means to someone who knows little statistics. 

(b) State and interpret the 95% confidence interval. 

(c) Interpret the confidence level. 

Step-by-Step Solution

Answer

a). On average the sample proportion will be within $3\%$ of the true proportion in 95% of all samples.

b). We are $95\%$ confident that the true population proportion is between $0.63$ and $0.69$.

c).  On average, in $95\%$ of all samples the corresponding confidence interval will contain the true population proportion $p$.

Part (a) Step 1: Given Information

The article gives the margin of error for a $95\%$ confidence level as $3$ percentage points.

Part (a) Step 2: Explanation

The margin of error is the amount of error that is made, on average, in estimating the population proportion $p$ by the sample proportion $\hat{p}$.

Thus on average the sample proportion will be within $3\%$ of the true proportion in $95\%$ of all samples.

Part (b) Step 1: Given Information

The article gives the margin of error for a $95\%$.

$\hat{p}=66\%=0.66$

$E=3\%=0.03$

Part (b) Step 2: Explanation

The boundaries of the confidence interval is the sample proportion increased/decreased by the margin of error:

$0.63=0.66-0.03=\hat{p}-E<p<\hat{p}+E=0.66+0.03=0.69$

We are $95\%$ confident that the true population proportion is between $0.63$ and $0.69$.

Part (c) Step 1: Given Information

The article gives the margin of error for a $95\%$ confidence level.

Part (c) Step 2: Explanation

The confidence level is $95\%$.

This means: On average, in $95\%$ of all samples the corresponding confidence interval will contain the true population proportion $p$.