A Statistics professor assigned 10 quizzes over the course of the semester. He wanted to see if there was a relationship between the total mark of all 10 quizzes and the final exam mark. There were 229 students who completed all the quizzes and wrote the final exam. The standard deviation of the total quiz marks was 13, and that of the final exam was 18. The correlation between the total quiz mark and the final exam was 0.7.
Based on the least squares regression line fitted to the data of the 229 students, if a student scored 23 points above the mean of total quiz marks, then how many points above the mean on the final would you predict her final exam grade to be? (Round your answer to one decimal place, but do not round in intermediate steps)
16.1 points
To predict the final exam grade based on the total quiz marks, we can use least squares regression line. The equation of the line is given by:
y = mx + b
Where y is the predicted final exam grade, x is the total quiz mark, m is the slope of the line, and b is the y-intercept.
To find the slope (m), we can use the formula:
m = r * (SD.y / SD.x)
Where r is the correlation coefficient between total quiz mark and final exam grade, SD.y is the standard deviation of the final exam grades, and SD.x is the standard deviation of the total quiz marks.
Given:
r = 0.7
SD.y = 18
SD.x = 13
Substituting the given values:
m = 0.7 * (18 / 13)
m ≈ 0.9615 (rounded to four decimal places)
Now, to find the y-intercept (b), we can use the formula:
b = mean.y - (m * mean.x)
Since we want to predict the final exam grade when the student scores 23 points above the mean of total quiz marks, we need to find the mean of the total quiz marks.
Given students who completed all 10 quizzes = 229
To find mean.x:
mean.x = sum of total quiz marks / number of students
mean.x = (sum of total quiz marks) / 229
Since we don't have the exact values of the total quiz marks, we cannot determine the exact mean.x to proceed with the calculation. Please provide more information or data regarding the sum of total quiz marks to continue solving the problem.
To find the predicted final exam grade, we need to use the least squares regression line equation. The equation is given by:
Y = a + b*X
Where Y represents the predicted final exam grade, X represents the total quiz mark, a represents the y-intercept, and b represents the slope.
We are given that the correlation between the total quiz mark and the final exam is 0.7. This means that the slope, b, can be calculated as:
b = correlation * (std deviation of final exam / std deviation of total quiz marks)
b = 0.7 * (18 / 13)
b = 0.7 * 1.3846
b ≈ 0.9692
Now, we need to find the y-intercept, a. We can use the formula:
a = mean of final exam - (b * mean of total quiz marks)
However, we are not given the mean of either the final exam or the total quiz marks. We can overcome this by considering that the mean of the final exam grade is equal to the mean of the predicted final exam grade, and the mean of the total quiz marks is equal to the mean of the actual total quiz marks.
This means that we need to find the mean of the actual total quiz marks. Since we are not given that value, we cannot calculate the exact predicted final exam grade. We can only calculate the change in the final exam grade compared to the mean.
Given that a student scored 23 points above the mean of the total quiz marks, we need to calculate how many points above the mean on the final exam grade this corresponds to. However, without knowing the mean values, we cannot perform this calculation.