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Chapter 7: Problem 98

Has trust in the legislative branch of government declined? A Gallup poll asked U.S. adults if they trusted the legislative branch of government in 2008 and again in 2017 . The results are shown in the table. $$ \begin{array}{|l|r|} \hline & \mathbf{2 0 0 8} & \mathbf{2 0 1 7} \\ \hline \text { Yes } & 399 & 358 \\ \hline \text { No } & 623 & 664 \\ \hline \text { Total } & 1022 & 1022 \\ \hline \end{array} $$ a. Find and compare the sample proportion for those who trusted the legislative branch in 2008 and in 2017 . b. Find the \(95 \%\) confidence interval for the difference in the population proportions. Assume the conditions for using the confidence interval are met. Based on the interval, has there been a change in the proportion of U.S. adults who trust the legislative branch? Explain.

Short Answer

Expert verified
The exact proportions and confidence interval depend on the calculations. However, the tendency can be noted that if the confidence interval contains zero, there hasn't likely been a significant change in trust. If all values in the interval are above zero, there may be a decrease in trust, while an interval below zero indicates an increase in trust.

Step by step solution

01

Find and compare the sample proportion

Calculate the sample proportions by taking the number of affirmative responses and dividing them by the total responses for each year. This suggests calculating \(p_{2008} = \frac{399}{1022}\) and \(p_{2017} = \frac{358}{1022}\) to find the percentage of respondents trusting the legislative branch in 2008 and 2017, respectively.
02

Find the 95% confidence interval

Once the sample proportions have been calculated, the 95% confidence interval for the difference of proportions can be calculated with the formula: \(\left(p_1 - p_2 \pm z \sqrt{\frac{(p_1 )(1 - p_1)}{n_1 }+\frac{(p_2 )(1 - p_2)}{n_2 }}\right)\). The z-value for a 95% confidence interval is approximately 1.96, \(p_1\) and \(p_2\) are the calculated proportions, and \(n_1\) and \(n_2\) are the total counts for each year.
03

Analyze the change

Once the confidence interval is determined, any noticeable change in the proportion of U.S. adults who trust the legislative branch may be noted. If the confidence interval contains zero, then there has likely been no significant change. If the entire interval is above zero, the proportion of trust might have decreased. And if the entire interval is below zero, the proportion might have increased.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportion Comparison
When evaluating changes in public opinion, comparing proportions from different time periods can provide valuable insights. In the given exercise, we assess whether trust in the legislative branch has declined over time by finding the sample proportions of trust in the years 2008 and 2017.

To find these proportions, we divide the number of responders who answered "Yes" by the total respondents for each year separately. For 2008, the calculation is \(p_{2008} = \frac{399}{1022}\), and similarly for 2017, \(p_{2017} = \frac{358}{1022}\). These calculations give the sample proportions of people trusting the legislative branch in those respective years.

The comparison of these proportions helps determine whether there is an apparent decrease, increase, or steadiness in public trust. A decrease in the proportion from 2008 to 2017 suggests a decline in trust.
Statistical Analysis
Statistical analysis enables us to make informed assertions about population parameters based on sample data. In this context, we delve into whether the shift in sample proportions represents a significant change in public trust towards the legislative branch.

To assist this process, we use a 95% confidence interval to estimate the difference in population proportions between the two years. This involves applying the formula: \[\left(p_1 - p_2 \pm z \sqrt{\frac{(p_1 )(1 - p_1)}{n_1 }+\frac{(p_2 )(1 - p_2)}{n_2 }}\right)\]where \(p_1\) and \(p_2\) are the proportions from 2008 and 2017, and \(n_1\) and \(n_2\) are the total sample sizes for those years. The \(z\)-value for a 95% confidence interval is approximately 1.96.

A confidence interval helps us understand the range within which the true difference in population proportions likely exists. If the interval does not include zero, it signifies a significant change in the proportion of public trust.
Public Trust Measurement
Measuring public trust is pivotal for assessing institutional credibility and democratic health. This exercise explores changes in public trust in the legislative branch, providing an example of how trust is quantified and assessed over time.

Through surveys, such as the Gallup poll mentioned, opinions are collected and analyzed. These surveys serve as a snapshot of public sentiment, reflecting broader societal perspectives.

The resultant data, transformed into proportions and analyzed through confidence intervals, offer a more scientific, evidence-based evaluation of perceived trust changes. This approach provides policy-makers and stakeholders with the necessary insights to intervene, if needed, to restore or enhance trust. Understanding trends and shifts in public trust helps inform decisions and measure the impact of legislative actions.

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Most popular questions from this chapter

Four women selected from a photo of 123 were found to have a sample mean height of 71 inches \((5\) feet 11 inches \()\). The population mean for all 123 women was \(64.6\) inches. Is this evidence that the sampling procedure was biased? Explain.

A random sample of likely voters showed that \(49 \%\) planned to support Measure \(\mathrm{X}\). The margin of error is 3 percentage points with a \(95 \%\) confidence level. a. Using a carefully worded sentence, report the \(95 \%\) confidence interval for the percentage of voters who plan to support Measure \(X\). b. Is there evidence that Measure X will fail? c. Suppose the survey was taken on the streets of Miami and the measure was a Florida statewide measure. Explain how that would affect your conclusion.

Is simple random sampling usually done with or without replacement?

Refer to Exercise \(7.77\) for information. This data set records results just for the boys. $$ \begin{array}{|lcc|} \hline & \text { Preschool } & \text { No Preschool } \\ \hline \text { Grad HS } & 16 & 21 \\ \hline \text { No Grad HS } & 16 & 18 \\ \hline \end{array} $$ a. Find and compare the percentages that graduated for each group, descriptively. Does this suggest that preschool was linked with a higher graduation rate? b. Verify that the conditions for a two-proportion confidence interval are satisfied. c. Indicate which one of the following statements is correct. i. The interval does not capture 0 , suggesting that it is plausible that the proportions are the same. ii. The interval does not capture 0 , suggesting that it is not plausible that the proportions are the same. iii. The interval captures 0 , suggesting that it is plausible that the population proportions are the same. iv. The interval captures 0 , suggesting that it is not plausible that the population proportions are the same. d. Would a \(99 \%\) confidence interval be wider or narrower?

According to a 2017 Pew Research survey, \(60 \%\) of young Americans aged 18 to 29 say the primary way they watch television is through streaming services on the Internet. Suppose a random sample of 200 Americans from this age group is selected. a. What percentage of the sample would we expect to watch television primarily through streaming services? b. Verify that the conditions for the Central Limit Theorem are met. c. Would it be surprising to find that 125 people in the sample watched television primarily through streaming services? Why or why not? d. Would it be surprising to find that more than \(74 \%\) of the sample watched television primarily through streaming services? Why or why not?
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