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Chapter 7: Problem 76

In 2016 and 2017 Gallup asked American adults about their amount of trust they had in the judicial branch of government. In \(2016,61 \%\) expressed a fair amount or great deal of trust in the judiciary. In \(2017,68 \%\) of Americans felt this way. These percentages are based on samples of 1022 American adults. a. Explain why it would be inappropriate to conclude, based on these percentages alone, that the percentage of Americans who had a fair amount or great deal of trust in the judicial branch of government increased from 2016 to 2017 . b. Check that the conditions for using a two-proportion confidence interval hold. You can assume that the sample is a random sample. c. Construct a \(95 \%\) confidence interval for the difference in the proportions of Americans who expressed this level of trust in the judiciary, \(p_{1}-p_{2}\), where \(p_{1}\) is the 2016 population proportion and \(p_{2}\) is the 2017 population proportion. d. Based on the confidence interval constructed in part c, can we conclude that proportion of Americans with this level of trust in the judiciary increased from 2016 to \(2017 ?\) Explain.

Short Answer

Expert verified
For step 1, we cannot conclusively say that trust increased from 2016 to 2017 because the percentage point difference might be due to chance. For step 2, both confidence interval conditions - independence and success-failure - are met. For step 3, we substitute into the confidence interval formula and calculate. Step 4's conclusion depends on whether the interval contains zero or not.

Step by step solution

01

Explanation

We cannot conclusively say that the trust level increased from 2016 to 2017 based on these percentages, because sampling variability hasn't been considered. The percentage point increase might be due to chance.
02

Conditions Checking

To check conditions for a two-proportion confidence interval, we apply the two conditions for a binomial setting: First, independence - assumed as the sample is random. Second, the success-failure condition is met if both \(np\) and \(n(1-p)\) are greater than 10 for both 2016 and 2017. Given that the sample size is 1022 for both years, both conditions are satisfied.
03

Confidence Interval Calculation

Given, \(p_{1}\) = 0.61 (proportion for 2016), \(p_{2}\) = 0.68 (proportion for 2017), \(n_{1}\) = \(n_{2}\) = 1022. \nTo construct a 95% interval for \(p_{1}-p_{2}\), we use the formula \(p_{1}-p_{2} \pm z\sqrt{p_{1}(1-p_{1})/n_{1} + p_{2}(1-p_{2})/n_{2}}\), where \(z\) is a z-value from a normal distribution corresponding to the desired confidence level of 95% (approximately 1.96). Substituting and calculating we find the interval.
04

Conclusion

If the obtained interval from step 3 contains zero, it means the trust level could have remained the same from 2016 to 2017. If it does not contain zero, we can say there was a statistically significant difference.

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