Several years ago, $34 \%$ of parents who had children in grades $\mathrm{K}-12$ were satisfied with the quality of education the students receive. $A$ recent poll asked 1,155 parents who have children in grades K-12 if they were satisfied with the quality of education the students receive. Of the 1,155 surveyed, 456 indicated that they were satisfied. Construct a $99 \%$ confidence interval to assess whether this represents evidence that parents' attitudes toward the quality of education have changed. What is the correct conclusion? A. Since the interval does not contain the proportion stated in the null hypothesis, there is insufficient evidence that parents' attitudes toward the quality of education have changed. B. Since the interval contains the proportion stated in the null hypothesis, there is insufficient evidence that parents' attitudes toward the quality of education have changed. C. Since the interval contains the proportion stated in the null hypothesis, there is sufficient evidence that parents' attitudes toward the quality of education have changed. D. Since the interval does not contain the proportion stated in the null hypothesis, there is sufficient evidence that parents' attitudes toward the quality of education
Solution
Step 1 ：First, calculate the proportion of parents who are satisfied with the quality of education in the recent poll. This is given by the number of parents who indicated they were satisfied divided by the total number of parents surveyed, which is \(\frac{456}{1155} = 0.395\).
Step 2 ：Next, construct a 99% confidence interval for this proportion. The formula for a confidence interval for a proportion is given by \(p \pm z \sqrt{\frac{p(1-p)}{n}}\), where p is the sample proportion, n is the sample size, and z is the z-score corresponding to the desired level of confidence. For a 99% confidence interval, the z-score is approximately 2.576.
Step 3 ：Substitute the values into the formula to get the confidence interval: \(0.395 \pm 2.576 \sqrt{\frac{0.395(1-0.395)}{1155}}\), which simplifies to \([0.358, 0.432]\).
Step 4 ：Finally, compare the confidence interval to the proportion of parents who were satisfied several years ago (0.34). The confidence interval does not contain 0.34, which was the proportion of parents who were satisfied several years ago.
Step 5 ：\(\boxed{\text{Therefore, we can conclude that parents' attitudes toward the quality of education have changed.}}\)
