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Chapter 7: Problem 62

A 2016 Pew Research poll found that \(61 \%\) of U.S. adults believe that organic produce is better for health than conventionally grown varieties. Assume the sample size was 1000 and that the conditions for using the CLT are met. a. Find and interpret a \(95 \%\) confidence interval for the proportion of U.S. adults to believe organic produce is better for health. b. Find and interpret an \(80 \%\) confidence interval for this population parameter. c. Which interval is wider? d. What happens to the width of a confidence interval as the confidence level decrease?

Short Answer

Expert verified
a. The calculated \(95 \%\) confidence interval indicates that we're \(95 \%\) confident that the actual proportion of U.S. adults who believe organic produce is healthier lies within that range. \n b. Similarly, the \(80 \%\) confidence interval interpretation means we're \(80 \%\) confident that the actual proportion lies within that range. c. The \(95 \%\) confidence interval is wider compared to the \(80 \%\) confidence interval. d. As the confidence level decreases, the width of the confidence interval also decreases.

Step by step solution

01

Find the 95% confidence interval

First, calculate the \(95 \%\) confidence interval. When using the Central Limit Theorem, the Z value for a \(95 \%\) confidence interval is approximately 1.96. The formula used is: \[CI = \overline{p} \pm Z \times \sqrt{\frac{\overline{p}(1- \overline{p})}{n}}\]where \(\overline{p}\) is the sample proportion, \(n\) is the sample size, and \(Z\) is the Z score corresponding to the desired confidence level. Substituting the given values, we have:\[CI = 0.61 \pm 1.96 \times \sqrt{\frac{0.61 \times 0.39}{1000}}\]
02

Find the 80% confidence interval

Applying a similar process, calculate the \(80 \%\) confidence interval. The Z value for an \(80 \%\) confidence interval is approximately 1.28. Using the same formula but substituting the appropriate Z-value, we have:\[CI = 0.61 \pm 1.28 \times \sqrt{\frac{0.61 \times 0.39}{1000}}\]
03

Compare Interval Widths

To determine which interval is wider, compare the range of values generated from the two previous steps. The range for the \(95 \%\) confidence interval will be from \(0.61 - 1.96 \times \sqrt{\frac{0.61 \times 0.39}{1000}}\) to \(0.61 + 1.96 \times \sqrt{\frac{0.61 \times 0.39}{1000}}\), and for the \(80 \%\) confidence interval this range will be from \(0.61 - 1.28 \times \sqrt{\frac{0.61 \times 0.39}{1000}}\) to \(0.61 + 1.28 \times \sqrt{\frac{0.61 \times 0.39}{1000}}\).
04

The Effect of Confidence Level on Interval Width

Conceptually, as the level of confidence decreases, the confidence interval becomes narrower. This is because a lower confidence level means that you're willing to accept a higher probability that the true population parameter falls outside your interval.

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