Question 1 (Multiple Choice Worth 5 points) ( 04.05 MC ) A parks and recreational board in Birch County is interested in estimating the proportion of its residents in favor of having more public parks in that county. A random sample of Birch County residents was selected. All the selected residents were asked, "Are you in favor of having more public parks in your county?" A $$ 98 \% $$ confidence interval for the proportion of residents in favor of having more public parks in that county was calculated to be $$ 0.54 \pm 0.03 $$. Which of the following statements is correct? At the $$ 98 \% $$ confidence level, the majority of area residents is in favor of having more public parks in that county. At the $$ 98 \% $$ confidence level, the estimate of 0.54 is within 0.03 of the true proportion of county residents in favor of having more public parks in that county. In repeated sampling, the true proportion of county residents in favor of having more public parks in that county will fall in the interval ( $$ 0.51,0.57 $$ ). In repeated sampling, $$ 98 \% $$ of the time the true proportion of county residents in favor of having more public parks in that county will be equal to 0.54 . Question $$ \mathbf{2} $$ (Multiple Choice Worth 5 points)

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1. The given interval is  
   $$
   0.54 \pm 0.03,
   $$
   which corresponds to the interval  
   $$
   (0.54-0.03, 0.54+0.03) = (0.51, 0.57).
   $$

2. This confidence interval means we are 98% confident that the true proportion $$ p $$ of residents in favor of more public parks lies within the interval (0.51, 0.57).

3. The statement that correctly interprets the confidence interval is:  
   „At the $$98\%$$ confidence level, the estimate of $$0.54$$ is within $$0.03$$ of the true proportion of county residents in favor of having more public parks in that county.“  
   This means that with 98% confidence, the maximum difference between the sample estimate $$0.54$$ and the true proportion is $$0.03$$.

4. The other options are incorrect because:
   - Option 1 implies that “the majority” is in favor, which is an inference but not the direct interpretation of the interval.
   - Option 3 misinterprets the meaning of repeated sampling; the true proportion is a fixed value and it is the computed interval from each sample that would capture the true proportion 98% of the time.
   - Option 4 incorrectly states that in 98% of repeated samples, the true proportion equals $$0.54$$, whereas $$0.54$$ is just the point estimate from the sample.

Thus, the correct choice is:

At the $$ 98\% $$ confidence level, the estimate of $$ 0.54 $$ is within $$ 0.03 $$ of the true proportion of county residents in favor of having more public parks in that county.
At the 98% confidence level, the estimate of 0.54 is within 0.03 of the true proportion of county residents in favor of having more public parks in that county.

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