Question

The IQ scores and English test scores of fifth grade students are given by the regression line Å· = -26.7 + 0.9346s, where Å· is the predicted English score and s is the IQ score. An actual English test score for a student is 65.7 with an IQ of 96. Find and interpret the residual.

1) -2.68; the regression line underpredicts the student's English test score.
2) 2.68; the regression line overpredicts the student's English test score.
3) -2.68; the regression line overpredicts the student's English test score.
4) 2.68; the regression line underpredicts the student's English test score.

Answer 1

The residual for the student's English test score is 2.68, indicating that the regression line underpredicted the student's score by this amount.

Step-by-step explanation:

The residual is the difference between the actual score and the predicted score from the regression line. To find the residual, you substitute the IQ score (s) of 96 into the regression equation Å· = -26.7 + 0.9346s to get the predicted English score. Then, you subtract this predicted score from the actual English test score of 65.7.

Predicted English score = -26.7 + 0.9346(96) = -26.7 + 89.7216 = 63.0216

Residual = Actual score - Predicted score = 65.7 - 63.0216 = 2.6784

The residual is 2.68, which means option 4 is correct: '2.68; the regression line underpredicts the student's English test score.' This indicates that the actual score was higher than the regression line predicted by 2.68 points.