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Has trust in the executive branch of government declined? A Gallup poll asked U.S. adults if they trusted the executive branch of government in 2008 and again in 2017 . The results are shown in the table. $$\begin{array}{|l|r|}\hline & \mathbf{2 0 0 8} & \mathbf{2 0 1 7} \\ \hline \text { Yes } & 623 & 460 \\\\\hline \text { No } & 399 & 562 \\\\\hline \text { Total } & 1022 & 1022 \\ \hline\end{array}$$ a. Find and compare the sample proportion for those who trusted the executive 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 executive branch? Explain.

Step by step solution

01

Compute Sample Proportions

The sample proportion is the number of successful outcomes divided by the total number of outcomes. From the table, the required proportions are calculated as follows: - For 2008: \( p_{2008} = \frac{623}{1022} \)- For 2017: \( p_{2017} = \frac{460}{1022} \)

02

Compare Sample Proportions

After finding the numerical values of these proportions, compare them to see which one is greater.

03

Compute Confidence Interval

A \(95\%\) confidence interval for the difference of two population proportions is given by the formula:\((p_1 - p_2) \pm Z * \sqrt{\frac{p_1*(1-p_1)}{n_1} + \frac{p_2*(1-p_2)}{n_2}}\) where \(Z = 1.96\) for a \(95\%\) confidence interval, \(p_1\) is the sample proportion for 2008, \(n_1\) is the sample size for 2008, \(p_2\) is the sample proportion for 2017, and \(n_2\) is the sample size for 2017. Plug in the values and calculate.

04

Interpret the Confidence Interval

The interpretation of confidence intervals in this context is that we are \(95\%\) confident that the true difference in proportions of U.S adults who trust the executive branch in 2008 and 2017 lies within this interval. If this interval includes zero, we cannot say with certainty that there is a difference in the proportions. If this interval does not include zero, there is a significant difference, and the direction of the difference can be determined by the sign of the lower and upper boundaries.

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Key Concepts

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

Sample Proportion

A sample proportion is a simple yet powerful concept in statistics. It helps us understand a part of a whole based on survey data. In the context of the Gallup poll exercise, the sample proportion is the ratio we obtain when dividing the number of people who answered "Yes" by the total respondents for a given year.

Think of it as a snapshot of opinions or behaviors within a group. In 2008, the sample proportion of those who trusted the executive branch was calculated as follows:

• Sample Proportion for 2008: \( p_{2008} = \frac{623}{1022} \)

By breaking it down, we see how many supported the executive branch out of the total surveyed in that year. The same calculation is repeated for 2017:

• Sample Proportion for 2017: \( p_{2017} = \frac{460}{1022} \)

This measure gives us a straightforward way to observe changes over time within the sample from year to year.

Population Proportion

While sample proportion looks at the survey data, the population proportion is the aim to measure trust levels across the entire U.S. adult population.
It's an estimate that the sample proportions provide. Keep in mind, the sample is a smaller representation of the population.

In hypothesis testing, we compare sample proportions to infer something about the population proportion, hoping the sample reflects the population accurately.
Based on sampling techniques, conditions should be met (like randomness of selection) to ensure that this reflection holds true.

The calculated confidence interval around the sample proportions gives insights about the range where the true population proportion might lie, indicating the trust level's real extent across all U.S. adults.

Hypothesis Testing

Hypothesis testing in this exercise evaluates whether there's a significant change in trust from 2008 to 2017 in the executive branch.
This process involves comparing sample proportions using formulated hypotheses.

The null hypothesis (

• \(H_0\)): states that there's no difference in the trust proportion between the two years.

The alternative hypothesis (

• \(H_a\)): proposes that a difference exists.

The heart of hypothesis testing lies in the confidence interval, which provides a range for the true difference of population proportions. A 95% confidence interval, for instance, means we'd expect the true difference to lie within the calculated interval 95 times out of 100 repeated polls.

If this interval does not include zero, it suggests a significant change in trust levels between the two years. However, if zero is within the interval, no significant change is evident, supporting the null hypothesis.