In a large class, there were 220 students who wrote both the midterm and the final exam. The standard deviation of the midterm grades was 14, and that of the final exam was 20. The correlation between the grades on the midterm and the final was 0.60.
Based on the least squares regression line fitted to the data of the 220 students, if a student scored 21 points below the mean on the midterm, then how many points below the mean on the final would you predict her final exam grade to be?
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To predict the final exam grade based on the midterm score, we can use the concept of linear regression. The least squares regression line can be found by using the formula:
Y = a + bX
Where Y is the predicted final exam grade, X is the midterm score, a is the intercept, and b is the slope.
To find the intercept (a) and slope (b), we need to use the given information. The correlation between the grades on the midterm and the final (r) is given as 0.60.
The formula for the slope (b) of the regression line is given by:
b = r * (σy / σx)
Where r is the correlation coefficient, σy is the standard deviation of final exam grades, and σx is the standard deviation of midterm grades.
In this case, r = 0.60, σy = 20 (standard deviation of final exam grades), and σx = 14 (standard deviation of midterm grades). Plugging in these values, we get:
b = 0.60 * (20 / 14)
Now, substitute the value of b into the equation to find a:
a = mean(y) - b * mean(x)
Assuming the means of midterm and final exam grades are zero (since we are dealing with differences from the mean), the equation simplifies to:
a = 0 - b * 0
Therefore, a = 0.
Now, we have the equation for the least squares regression line:
Y = bX
To predict the final exam grade if a student scored 21 points below the mean on the midterm, we substitute X = -21 into the equation:
Y = b * (-21)
Now, we can calculate the predicted final exam grade:
Y = (0.60 * (20 / 14)) * (-21)
Simplifying, we get:
Y ≈ -18.86
Therefore, the predicted final exam grade for a student who scored 21 points below the mean on the midterm would be approximately 18.86 points below the mean on the final exam.