In a course there is a midterm exam, worth 40% of the course grade, and a final
exam worth 60%. On the midterm the mean grade was 80 with a standard deviation of
10, and on the final the mean was 75 with a standard deviation of 15. The correlation
between the two exams is 0.6 .
Find the expected course grade, and the standard deviation of the course grade
To find the expected course grade, we need to calculate the weighted average of the midterm and final exam grades.
Step 1: Calculate the expected scores for the midterm and final exams.
The mean score for the midterm is 80, and the mean score for the final is 75.
Expected score on the midterm = 80
Expected score on the final = 75
Step 2: Calculate the weighted average of the midterm and final exam grades.
Since the midterm is worth 40% and the final exam is worth 60%, we need to multiply the expected scores by their respective weights.
Weighted average = (Midterm score * Weight of midterm) + (Final score * Weight of final)
= (80 * 0.4) + (75 * 0.6)
= 32 + 45
= 77
Therefore, the expected course grade is 77.
To calculate the standard deviation of the course grade, we need to use the formula for combining variances of independent variables.
Step 1: Calculate the variance of the midterm and final exam grades.
The standard deviation of the midterm is 10, and the standard deviation of the final is 15.
Variance of midterm = (Standard deviation of midterm)^2
= 10^2
= 100
Variance of final = (Standard deviation of final)^2
= 15^2
= 225
Step 2: Calculate the combined variance of the midterm and final exam grades using the correlation coefficient.
Combined variance = (Variance of midterm * Weight of midterm ^ 2) + (Variance of final * Weight of final ^ 2) + (2 * Correlation * Standard deviation of midterm * Weight of midterm * Standard deviation of final * Weight of final)
= (100 * 0.4^2) + (225 * 0.6^2) + (2 * 0.6 * 10 * 0.4 * 15 * 0.6)
= 16 + 81 + 64.8
= 161.8
Step 3: Calculate the standard deviation of the course grade by taking the square root of the combined variance.
Standard deviation of course grade = √(Combined variance)
= √(161.8)
≈ 12.72
Therefore, the standard deviation of the course grade is approximately 12.72.