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Elapsed Time to Earn a Bachelor's Degree A researcher with the Department of Education followed a cohort of students who graduated from high school in \(1992,\) monitoring the progress the students made toward completing a bachelor's degree. One aspect of his research was to determine whether students who first attended community college took longer to attain a bachelor's degree than those who immediately attended and remained at a 4-year institution. The data in the table summarize the results of his study. $$ \begin{array}{lcc} & \begin{array}{l} \text { Community College } \\ \text { to Four-Year Transfer } \end{array} & \text { No Transfer } \\ \hline n & 268 & 1145 \\ \hline \begin{array}{l} \text { Sample mean time to } \\ \text { graduate, in years } \end{array} & 4.43 \\ \hline \begin{array}{l} \text { Sample standard } \\ \text { deviation time to } \\ \text { graduate, in years } \end{array} & 1.162 & 1.015 \\ \hline \end{array} $$ (a) What is the response variable in this study? What is the explanatory variable? (b) Explain why this study can be analyzed using the methods of this section. (c) Does the evidence suggest that community college transfer students take longer to attain a bachelor's degree? Use an \(\alpha=0.01\) level of significance. (d) Construct a \(95 \%\) confidence interval for \(\mu\) community college \(\mu_{\text {no transfer }}\) to approximate the mean additional time it takes to complete a bachelor's degree if you begin in community college. (e) Do the results of parts (c) and (d) imply that community college causes you to take extra time to earn a bachelor's degree? Cite some reasons that you think might contribute to the extra time to graduate.

Step by step solution

01

Identify the Response and Explanatory Variables

The response variable is the 'time to graduate with a bachelor's degree.' The explanatory variable is 'the type of institution first attended' (community college or 4-year institution).

02

Suitability of Methods

This study can be analyzed using the methods of comparing two means because it is comparing the average time to graduate for two different groups: community college transfers and those who did not transfer.

03

Hypothesis Testing

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Key Concepts

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

Response Variable

The response variable in a study is the outcome or the main variable of interest that the researcher is trying to measure or predict. In the context of the study regarding the time to earn a bachelor's degree, the response variable is the 'time to graduate with a bachelor's degree.'
This variable quantifies the amount of time, in years, students take to complete their bachelor's degree.
It is important because it provides valuable insights into the duration of undergraduate education, which can be influenced by various factors, such as the type of institution attended. Understanding the response variable allows researchers to focus on what they aim to analyze and provides a basis for comparing different groups.

Explanatory Variable

The explanatory variable, also known as the independent variable, is the factor that is being manipulated or categorized to observe its effect on the response variable.
In this study, the explanatory variable is 'the type of institution first attended,' which can be either a community college or a 4-year institution.
This variable helps determine if the initial type of educational institution has any impact on the time it takes to graduate with a bachelor's degree. By separating students based on this criterion, researchers can compare the average time to graduate between the two groups and explore potential causative relationships.

Comparing Two Means

Comparing two means involves statistical methods used to determine if there is a significant difference between the average values of two groups.
In this study, we compare the mean time to graduate between students who transferred from community colleges and those who did not transfer.
The sample mean and standard deviation for each group provide a basis for this comparison.
We can use methods such as t-tests to evaluate whether the observed difference between the sample means is statistically significant, which helps assess if community college transfer students take longer to attain a bachelor's degree than those who remain at a 4-year institution.

Hypothesis Testing

Hypothesis testing is a statistical technique used to make inferences about a population based on sample data.
In the context of this study, we perform a hypothesis test to determine if there is enough evidence to suggest that community college transfer students take longer to attain a bachelor's degree than students who did not transfer.
We set up the null hypothesis (\boldmath\(H_0\)) stating that there is no difference in mean graduation time between the two groups. Our alternative hypothesis (\boldmath\(H_a\)) states that community college transfer students take longer (i.e., their mean graduation time is higher).
Using an \boldmath\(\boldmath{\balaphadef\boldsymbol(\balpha)=0.01\boldmath)\) level of significance, we calculate the test statistic and its corresponding p-value to assess whether to reject the null hypothesis.

Confidence Interval

A confidence interval provides a range of values within which the true population parameter is likely to fall.
In this study, we construct a 95% confidence interval for the difference in mean graduation times between community college transfer students and non-transfer students.
The confidence interval helps estimate the mean additional time it takes to complete a bachelor's degree if one begins at a community college.
If the interval does not include 0, it suggests a significant difference in graduation times.
This method offers a way to gauge the precision of our estimate and the potential impact of attending a community college on graduation time.