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You Explain It! Study Time and Exam Scores After the first exam in a statistics course, Professor Katula surveyed 14 randomly selected students to determine the relation between the amount of time they spent studying for the exam and exam score. She found that a linear relation exists between the two variables. The least-squares regression line that describes this relation is \(\hat{y}=6.3333 x+53.0298\). (a) Predict the exam score of a student who studied 2 hours. (b) Interpret the slope. (c) What is the mean score of students who did not study? (d) A student who studied 5 hours for the exam scored 81 on the exam. Is this student's exam score above or below average among all students who studied 5 hours?

Step by step solution

01

- Predict score for 2 study hours

Use the given regression line \(\hat{y}=6.3333x+53.0298\) to find the predicted exam score for a student who studied for 2 hours. Substituting x=2 into the equation: \hat{y}=6.3333(2)+53.0298.\ Calculate the result.

02

- Calculate the prediction

Substitute x with 2: \(\hat{y}=6.3333 \times 2+53.0298=12.6666+53.0298=65.6964\). So, the predicted score is 65.6964.

03

- Interpret the slope

The slope of the regression line is 6.3333. This means that for every additional hour of study, the exam score is predicted to increase by 6.3333 points.

04

- Calculate mean score for zero study hours

Use the regression equation to find the mean score for students who did not study. Substitute x=0 into the equation: \(\hat{y}=6.3333 \times 0 + 53.0298=53.0298\). The mean score for students who did not study is 53.0298.

05

- Compare score of student who studied 5 hours

Predict the score for a student who studied 5 hours: substitute x=5 into the regression equation: \(\hat{y} =6.3333 \times 5 + 53.0298 =31.6665+53.0298=84.6963\). Since the actual score of the student who studied 5 hours is 81, which is lower than the predicted score of 84.6963, this score is below average among all students who studied 5 hours.

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Key Concepts

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

least-squares regression

Least-squares regression is a method to find the best-fitting straight line through a set of points. This line, called the regression line, minimizes the sum of the squares of the vertical distances of the points from the line. In our example, the regression line is given by the equation \(\[\begin{equation} \hat{y} = 6.3333x + 53.0298 \end{equation}\]\). This means that for any given value of x (study hours), you can predict y (exam score) using this equation. The least-squares method ensures that the predicted values are as close as possible to the actual values of the dependent variable.

predictive modeling

Predictive modeling involves using statistical techniques, like linear regression, to create a model that predicts future outcomes based on past data. Using the regression line \(\[\begin{equation} \hat{y} = 6.3333x + 53.0298 \end{equation}\]\), we can predict the exam score of a student who studied for any number of hours. For instance, to predict the score of a student who studied for 2 hours, we substitute x with 2 in the equation: \(\[\begin{equation} \hat{y} = 6.3333 \times 2 + 53.0298 = 65.6964 \end{equation}\]\). Thus, the predicted exam score is approximately 65.7. This model provides valuable insights and can be used for academic planning and intervention.

interpretation of slope

The slope of a regression line is crucial as it explains the relationship between the independent and dependent variables. In the equation \(\[\begin{equation} \hat{y} = 6.3333x + 53.0298 \end{equation}\]\), the slope is 6.3333. This means that for every additional hour of study, a student's exam score is expected to increase by 6.3333 points. To put it simply, more study hours lead to higher scores. Understanding this concept allows students and educators to gauge the effectiveness of study time on performance.

linear relationships

A linear relationship between two variables means that the change in one variable is consistent with a change in the other variable. In the context of the professor's study, there is a linear relationship between study hours and exam scores. The straight-line equation \(\[\begin{equation} \hat{y} = 6.3333x + 53.0298 \end{equation}\]\) captures this relationship, indicating that study time directly impacts exam scores in a predictable manner. Knowing there is a linear relationship helps in forming strategies based on quantifiable data. For example, educators can recommend specific study hours aimed at achieving target exam scores.