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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

Calculate the proportions

Proportion is the ratio of the part to the whole, so to calculate the proportion of those who trusted the executive branch in 2008 and 2017, divide the 'Yes' responses by the Total responses for each year.\(p_{2008} = \frac{623}{1022}\)\(p_{2017} = \frac{460}{1022}\)

02

Calculate the double proportion standard error

The formula for the double proportion standard error is \(\sqrt{\frac{p*(1-p)}{n}}\) where p is the proportion and n is the total sample size.For 2008, \(SE_{2008} = \sqrt{\frac{p_{2008}*(1-p_{2008})}{1022}}\)And for 2017, \(SE_{2017} = \sqrt{\frac{p_{2017}*(1-p_{2017})}{1022}}\)

03

Calculate the standard error for the mean difference

This is computed by the formula \(\sqrt{SE^2_{2008} + SE^2_{2017}}\)

04

Get the Z value

A 95% confidence level commonly associates with a Z-score of 1.96

05

Find the Confidence Interval

A Confidence Interval is a range of values derived from a data set, calculated from the given Z value and standard error.The formula for the Confidence Interval is CI = \(p_{2008} - p_{2017} ± (Z*SE)\)

06

Interpret the Results

Based on the computed Confidence Interval, decide if there is a significant difference in the population proportions. If the confidence interval does not contain 0, then there is a significant change in the proportion of U.S. adults who trust the executive branch.

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Key Concepts

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

Confidence Interval

When analyzing statistical data, a confidence interval gives a range where we expect the true population parameter to lie, within a certain level of certainty. It is based on the data obtained from a sample and provides upper and lower bounds for the estimations.

For instance, in the context of the Gallup poll, a confidence interval can tell us with 95% confidence the range in which the true difference in the proportion of U.S. adults who trusted the executive branch in 2008 compared to 2017 lies. To calculate this, we use the sample data and a Z-score, which corresponds to the desired level of confidence (1.96 for 95%). The wider the interval, the less precise is our estimate, but the more confident we can be that it contains the true difference in population proportions.

Proportion Calculation

Calculating a proportion is an essential skill in statistics, particularly when dealing with population studies. It represents the part of the population that has a particular characteristic, calculated as the ratio of the number in that group to the total population being studied.

In the Gallup poll example, the proportion of individuals who trusted the executive branch in a given year is found by dividing the number of 'Yes' responses by the total number of responses. For detailed and solid conclusions, it's crucial to compare these proportions carefully between the two years - an exercise that showcases the potential changes in public opinion over time.

Standard Error

The standard error reflects the variability of a sample statistic, such as the mean or a proportion, from sample to sample. It measures how precisely the sample estimates the population parameter.

In our Gallup poll exercise, we calculate the standard error of each proportion separately to help estimate the standard error of the difference between them. The formula involves both the proportion of 'Yes' answers and the sample size. With this information, we compute the standard error of the mean difference, combining the individual standard errors. Understanding the standard error is key to interpreting the confidence interval, as it influences the interval's width and, consequently, the certainty we have regarding the population proportion.