Question

11.6 A mathematics placement test is given to all entering freshmen at a small college. A

student who receives a grade below 35 is denied admission to the regular mathematics
course and placed in a remedial class. The placement test scores and the final grades for
20 students who took the regular course were recorded as follows:
Placement Test : 50 35 35 40 55 65 35 60 90 35 90 80 60 60 60 40 55 50 65 50
Course Grade : 53 41 61 56 68 36 11 70 79 59 54 91 48 71 71 47 53 68 57 79
(a) Plot a scatter diagram.
(b) Find the equation of the regression line to predict course grades from placement test scores.
(c) Graph the line on the scatter diagram.
(d) If 60 is the minimum passing grade, below which placement test score should students in the future be denied admission to this course?

Answer 1

To solve this problem, we can use linear regression to find the equation of the regression line that predicts course grades from placement test scores. Let's go through the steps:

(a) Plotting the scatter diagram:

Placement Test (x-axis): 50 35 35 40 55 65 35 60 90 35 90 80 60 60 60 40 55 50 65 50

Course Grade (y-axis): 53 41 61 56 68 36 11 70 79 59 54 91 48 71 71 47 53 68 57 79

We can plot these points on a graph to visualize the data.

(b) Finding the equation of the regression line:

We can use the least squares method to find the equation of the regression line.

Step 1: Calculate the means of the placement test scores (x) and course grades (y):

mean_x = (50 + 35 + 35 + 40 + 55 + 65 + 35 + 60 + 90 + 35 + 90 + 80 + 60 + 60 + 60 + 40 + 55 + 50 + 65 + 50) / 20 = 55

mean_y = (53 + 41 + 61 + 56 + 68 + 36 + 11 + 70 + 79 + 59 + 54 + 91 + 48 + 71 + 71 + 47 + 53 + 68 + 57 + 79) / 20 = 57.9

Step 2: Calculate the deviations from the means for each data point:

(x - mean_x) and (y - mean_y)

-5 -20 -20 -15 0 10 -20 5 35 -20 35 25 5 5 5 -15 0 -5 10 -5

-4 -17 3 -1 10 -21 -46 12 21 1 -3 33 -9 13 13 -10 -4 10 -0 21

Step 3: Calculate the product of the deviations:

(x - mean_x) * (y - mean_y)

20 340 -60 15 0 -210 920 60 735 -20 -105 825 -45 65 65 150 0 -50 0 105

Step 4: Calculate the squared deviations of x:

(x - mean_x)^2

25 400 400 225 0 100 400 25 1225 400 1225 625 25 25 25 225 0 25 100 25

Step 5: Calculate the equation of the regression line:

The slope (b) of the regression line is given by:

b = Σ((x - mean_x) * (y - mean_y)) / Σ((x - mean_x)^2)

b = (20 + 340 - 60 + 15 + 0 - 210 + 920 + 60 + 735 - 20 - 105 + 825 - 45 + 65 + 65 + 150 + 0 - 50 + 0 + 105) / (25 + 400 + 400 + 225 + 0 + 100 + 400 + 25 + 1225 + 400 + 1225 + 625 + 25 + 25 + 25 + 225 + 0 + 25 + 100 + 25)

b = 4115 / 10275 ≈ 0.400

The y-intercept (a) of the regression line can be calculated as:

a = mean_y - b * mean_x

a = 57.9 - 0.400 * 55 ≈ 37.9

Therefore, the equation of the regression line is:

Course Grade = 0.400 * Placement Test + 37.9

(c) Graphing the line on the scatter diagram:

We can plot the regression line on the scatter diagram from part (a) to visualize the relationship between the placement test scores and course grades.

(d) Finding the placement test score for the minimum passing grade:

If 60 is the minimum passing grade, we need to find the corresponding placement test score using the equation of the regression line:

60 = 0.400 * Placement Test + 37.9

Subtracting 37.9 from both sides:

22.1 = 0.400 * Placement Test

Dividing both sides by 0.400:

Placement Test = 22.1 / 0.400 ≈ 55.25

Therefore, students in the future should be denied admission to this course if their placement test score is below approximately 55.25.