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
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?

Verified step by step guidance

1. Identify the given regression equation: \(\hat{y} = 6.3333x + 53.0298\), where \(x\) is the number of study hours and \(\hat{y}\) is the predicted exam score.
2. Substitute the given study time \(x = 5\) hours into the regression equation to find the predicted (average) exam score for students who studied 5 hours: \(\hat{y} = 6.3333 \times 5 + 53.0298\).
3. Calculate the predicted exam score \(\hat{y}\) (do not compute the final value here, just set up the expression). This value represents the average exam score for students who studied 5 hours.
4. Compare the actual exam score of the student, which is 81, to the predicted average score \(\hat{y}\) for 5 hours of study.
5. If the actual score (81) is greater than the predicted score \(\hat{y}\), then the student's score is above average; if it is less, then the score is below average.

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 hours.

Predicted Value (ŷ) and Interpretation

The predicted value ŷ is the exam score estimated by the regression line for a given study time x. Comparing an actual score to ŷ helps determine if the score is above or below average for that study time, indicating whether the student performed better or worse than expected.

Residuals and Their Meaning

A residual is the difference between the observed exam score and the predicted score from the regression line. A positive residual means the student scored above average for their study time, while a negative residual indicates a below-average score, helping assess individual performance relative to the model.