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You would like to determine what students at your school would be willing to do to help address global warming and the development of alternatively fueled vehicles. To do this, you take a random sample of 100 students. One question you ask them is, "How high of a tax would you be willing to add to gasoline (per gallon) in order to encourage drivers to drive less or to drive more fuel-efficient cars?" You also ask, "Do you believe (yes or no) that global warming is a serious issue that requires immediate action such as the development of alternatively fueled vehicles?" In your statistical analysis, use inferential methods to compare the mean response on gasoline taxes (the first question) for those who answer yes and for those who answer no to the second question. For this analysis, a. Identify the response variable and the explanatory variable. b. Are the two groups being compared independent samples or dependent samples? Why? c. Identify a confidence interval you could form to compare the groups, specifying the parameters used in the comparison.

Step by step solution

01

Identify Variables

First, we identify the response and explanatory variables. The response variable is the amount of tax students are willing to add to gasoline (per gallon), as it is the main outcome we are interested in measuring. The explanatory variable is whether the student believes global warming requires immediate action, as this might influence their willingness to pay a higher tax.

02

Determine Sample Dependency

The two groups, students who believe global warming requires immediate action and those who do not, are independent samples. The reason is that the opinion of one student does not affect another's opinion; each student's response is separate and unrelated to others.

03

Select Confidence Interval

For comparing the mean tax willingness between two independent groups, we can use a confidence interval for the difference between two means. Assuming normal distribution, the formula for the confidence interval is \( \bar{x}_1 - \bar{x}_2 \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \), where \( \bar{x}_1 \) and \( \bar{x}_2 \) are the sample means, \( s_1 \) and \( s_2 \) are the sample standard deviations, \( n_1 \) and \( n_2 \) are the sample sizes, and \( t^* \) is the t-score from the t-distribution that corresponds to the desired confidence level.

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Key Concepts

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

Response Variable

In inferential statistics, the response variable is a critical concept. It serves as the primary outcome or measure of interest in your study. In the scenario described, the response variable is the amount of tax that students are willing to add to gasoline per gallon.

This variable represents what you are trying to understand or predict within the scope of your analysis. By analyzing this response, you can infer broader conclusions regarding students’ willingness to support actions addressing global issues like global warming. Understanding the response variable helps you gauge the effectiveness of potential environmental policies or interventions, such as fuel taxes, in altering behavior.

Explanatory Variable

An explanatory variable is one that you think might influence or affect your response variable. In your study, the explanatory variable is whether students believe global warming requires immediate action.

* This variable is categorical, with two possible values: 'yes' or 'no'.
* It helps you explore why students might be willing to support higher gasoline taxes.

By identifying this variable, you can attempt to uncover relationships between students’ beliefs and their willingness to pay more for gasoline. The explanatory variable serves as a lens to interpret human behaviors and attitudes in response to important environmental issues.

Confidence Interval

A confidence interval provides a range of values that are believed to encompass a population parameter with a certain level of confidence. When comparing two groups, like students who believe global warming requires immediate action versus those who do not, a confidence interval can help determine the difference in their average willingness to pay an additional gasoline tax.

The formula used, given a normal distribution assumption, is crucial: \[ \bar{x}_1 - \bar{x}_2 \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \]

• \( \bar{x}_1 \) and \( \bar{x}_2 \) represent the mean responses for each group.
• \( s_1 \) and \( s_2 \) are the standard deviations of the responses.
• \( n_1 \) and \( n_2 \) stand for the number of observations in each group.
• \( t^* \) is the t-score associated with your chosen confidence level, often 95%.

A confidence interval not only expresses the degree of uncertainty around the estimate but also provides a way to assess the statistical significance of observed differences between group means.

Independent Samples

Understanding whether samples are independent is critical in statistical analysis. In your study to explore students' opinions on gasoline taxes, the two groups (those who agree that global warming requires immediate action and those who do not) evolve independently.

* Each student's response is unique to that student, unaffected by the opinions of others.
* This independence means there is no intrinsic link or pairing between respondents in the two groups.

Distinguishing independent samples is essential because it determines the type of statistical tests you can use. In this case, methods suited to independent samples, like the two-sample t-test, are appropriate. Such understanding helps ensure the validity of your conclusions about students' attitudes towards global warming and related financial responsibilities.