91Ó°ÊÓ

A Pew Research foundation poll indicates that among 1,099 college graduates, \(33 \%\) watch The Daily Show. Meanwhile, \(22 \%\) of the 1,110 people with a high school degree but no college degree in the poll watch The Daily Show. A \(95 \%\) confidence interval for \(\left(p_{\text {college grad }}-p_{\mathrm{HS} \text { or less }}\right),\) where \(p\) is the proportion of those who watch The Daily Show, is \((0.07,0.15) .\) Based on this information, determine if the following statements are true or false, and explain your reasoning if you identify the statement as false. \({ }^{29}\) (a) At the \(5 \%\) significance level, the data provide convincing evidence of a difference between the proportions of college graduates and those with a high school degree or less who watch The Daily Show. (b) We are \(95 \%\) confident that \(7 \%\) less to \(15 \%\) more college graduates watch The Daily Show than those with a high school degree or less. (c) \(95 \%\) of random samples of 1,099 college graduates and 1,110 people with a high school degree or less will yield differences in sample proportions between \(7 \%\) and \(15 \%\). (d) A \(90 \%\) confidence interval for \(\left(p_{\text {college grad }}-p_{\text {HS or less }}\right)\) would be wider. (e) A \(95 \%\) confidence interval for \(\left(p_{\mathrm{HS}}\right.\) or less \(\left.-p_{\text {college grad }}\right)\) is (-0.15,-0.07) .

Step by step solution

01

Statement (a) Analysis

The confidence interval is (0.07, 0.15), which does not include zero. At the \( 5\% \) significance level, a confidence interval that doesn't include zero indicates statistically significant evidence of a difference. Therefore, Statement (a) is true.

02

Statement (b) Analysis

The confidence interval (0.07, 0.15) means that the difference in proportions is estimated to be between 7% more to 15% more college graduates watching The Daily Show than those with a high school degree or less. However, the statement says "7% less to 15% more," which is incorrect. So, Statement (b) is false.

03

Statement (c) Analysis

A confidence interval estimates the range within which the actual population parameter is likely to lie, not each sample. Statement (c) incorrectly describes the interpretation of a confidence interval by applying it to random samples, not the parameter. Therefore, Statement (c) is false.

04

Statement (d) Analysis

A \( 90\% \) confidence interval gives us less certainty about the estimate, meaning it would be narrower, not wider. A wider interval would result from a higher confidence level, such as \( 99\% \). Thus, Statement (d) is false.

05

Statement (e) Analysis

The confidence interval for \( p_{\text{HS or less}} - p_{\text{college grad}} \) is the reverse order of the given interval \( (0.07, 0.15) \), resulting in \( (-0.15, -0.07) \). So, Statement (e) is true.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a crucial process in statistical analysis. It helps us determine if there is enough evidence to support a specific claim about a population parameter. In this exercise, we are interested in whether there's a difference in the proportions of college graduates versus non-college graduates who watch The Daily Show. We begin with the null hypothesis, which states that there is no difference between the two groups' proportions, and the alternative hypothesis, which posits there is a difference. In this situation, a significance level of 5% (\( \alpha = 0.05 \)) is used to test our hypothesis.

• If the confidence interval does not include zero, as seen here (0.07, 0.15), it suggests rejecting the null hypothesis.
• This implies that the observed difference is statistically significant, allowing us to assert that the difference in viewership is not due to chance.

The confidence interval provides a range of probable differences between the two proportions, reinforcing our hypothesis test findings.

Proportions

When we talk about proportions in statistics, we are examining parts of a whole in terms of percentages. Here, the exercise deals with the proportions of distinct groups who watch The Daily Show. For college graduates, the proportion is given as 33%, while non-college graduates have a proportion of 22%. These values represent the parts of each group who are viewers. Understanding these proportions:

• The difference in proportions helps us assess the relative likelihood of viewing behaviors between the groups.
• The estimated confidence interval, (0.07, 0.15), provides the likely range of actual differences in these proportions at a 95% confidence level.

The statistical method of using proportions is powerful when comparing different groups as it accounts for each group's size, ensuring the comparison is fair and comprehensive.

Statistical Significance

Statistical significance is about judging whether the results of our analysis are likely to be genuine or if they occurred by chance. In this exercise, the idea revolves around whether the difference between college graduates and non-college graduates watching The Daily Show is statistically significant. Considerations in determining significance:

• Since the confidence interval of (0.07, 0.15) does not include zero, we have statistically significant evidence that the difference is real.
• A significance level of 5% means we are willing to accept a 5% chance of incorrectly rejecting the null hypothesis.
• This finding suggests that the likelihood of the observed difference occurring by chance is low.

Statistical significance lends credibility to our results, indicating that the differences in the sample reflect true differences in the population.

Survey Analysis

Survey analysis helps us understand population characteristics based on survey data. This process involves gathering responses from different groups to make informed inferences about those groups. For this exercise, a survey was conducted with college graduates and non-college graduates to determine their preferences for watching The Daily Show. Key aspects of survey analysis include:

• Sampling: The survey involved two distinct groups, capturing representative samples from both college graduates and non-college graduates.
• Data interpretation: Through analyzing the responses, we can assess differences or trends in viewing preferences among the groups.
• Confidence intervals: These provide a quantified uncertainty measure in our estimates, helping us draw conclusions confidently.

By conducting survey analysis, we can gain insights into how different educational backgrounds might influence media consumption habits, contributing to broader understandings of societal behavior.