Final answer:
The correlation coefficient of 0.567 shows a moderate positive association between test preparation time and test scores. The LSRL equation can be calculated to predict test scores based on preparation time. Anna's residual will indicate whether she scored higher or lower than the LSRL would predict for her 4 hours of test preparation.
Step-by-step explanation:
Understanding the Relationship Between Test Prep Time and Test Scores
Sophia is analyzing the association between test preparation time and test scores among a stratified sample of 27 students. She found the mean test preparation time to be 2.423 hours with a standard deviation of 0.676 hours. The mean test score is 79.33% with a standard deviation of 8.684%. The correlation coefficient is 0.567.
a. Strength and Direction of the Association
The correlation coefficient of 0.567 indicates a moderate positive association between test preparation time and test scores. This means as study time increases, test scores tend to increase as well.
b. Least Squares Regression Line (LSRL)
The equation of the LSRL can be calculated using the given statistics, and it will take the form of y = mx + b, where m represents the slope and b the y-intercept. The slope indicates the change in test score for each additional hour of study, whereas the y-intercept represents the expected test score for someone who did not spend any time preparing.
c. Residual Calculation for Anna
Anna's residual is the difference between her actual test score and the score predicted by the LSRL for her preparation time of 4 hours. If her actual score is higher than the predicted one, the residual will be positive, indicating that she scored better than expected based on her preparation time.1