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:
ŷ = 6.3333x + 53.0298
c. What is the mean score of students who did not study?

Verified step by step guidance

1. Identify the variables in the regression equation: here, \(x\) represents the amount of time spent studying, and \(\hat{y}\) represents the predicted exam score.
2. Understand that the mean score of students who did not study corresponds to the predicted exam score when the study time \(x = 0\).
3. Substitute \(x = 0\) into the regression equation \(\hat{y} = 6.3333x + 53.0298\) to find the predicted score for students who did not study.
4. Calculate \(\hat{y} = 6.3333 \times 0 + 53.0298\) to get the mean exam score for students with zero study time.
5. Interpret the result as the estimated average exam score for students who did not spend any time studying, based on the regression model.

Key Concepts

Least-Squares Regression Line

The least-squares regression line is a straight line that best fits the data points by minimizing the sum of the squared differences between observed and predicted values. It models the relationship between an independent variable (study time) and a dependent variable (exam score), allowing predictions of exam scores based on study time.

Interpretation of the Regression Equation

The regression equation ŷ = 6.3333x + 53.0298 expresses the predicted exam score (ŷ) as a function of study time (x). The slope (6.3333) indicates the expected increase in score for each additional hour studied, while the intercept (53.0298) represents the predicted exam score when study time is zero.

Mean Score for Zero Study Time

To find the mean exam score for students who did not study, substitute x = 0 into the regression equation. This yields the intercept value, which estimates the average exam score for students with zero study hours, reflecting the baseline performance without studying.