The prediction for these students was off by more than 12 points.
To find the prediction error for 1/3 of the students, we'll need to use the formula for the standard error of estimate (SEE) in a linear regression.
The formula for SEE is:
SEE=SD_y × √1−r^2
Where:
SDy is the standard deviation of the final scores (15 in this case).r is the correlation coefficient between midterm and final scores.
You've mentioned that the correlation between the midterm and final scores is given but was cut off in your message. Assuming the correlation is r=0.6 (just as an example), let's proceed with this calculation:
SEE=15× √1−0.6^2
SEE=15× √1−0.36
SEE=15× √0.64
SEE=15×0.8
SEE=12
So, the standard error of estimate is 12 points.
To find out the prediction error for 1/3 of the students, we'll look at the standard deviation, which covers about 68% of the data in a normal distribution. Since we're dealing with prediction errors for 1/3 of the students, which is beyond 68% of the data (i.e., beyond one standard deviation), we'll use the standard error of estimate (12 points) as a reference for the prediction error.
For 1/3 of the students, the prediction was off by more than one standard deviation (12 points). Therefore, the prediction for these students was off by more than 12 points.