This data shows the final exam scores of 10 randomly selected students and the number of hours they studied for an exam. (hours, scores) (3, 65), (5, 80), (2, 60), (8, 88), (2, 66), (4, 78), (4, 85), (5, 90), (6, 80), (3, 71)
Find the equation of the regression line for the given data. (Points : 8)
y = 4.4747x + 57.506
y = -4.4747x + 57.506
y = 57.506x + 4.4747
y = 57.506x – 4.4747
To find the equation of the regression line for the given data, you can follow these steps:
Step 1: Calculate the mean (average) of both the hours studied and the scores.
For the hours studied:
Mean(hours) = (3 + 5 + 2 + 8 + 2 + 4 + 4 + 5 + 6 + 3) / 10 = 4.2
For the scores:
Mean(scores) = (65 + 80 + 60 + 88 + 66 + 78 + 85 + 90 + 80 + 71) / 10 = 76.3
Step 2: Calculate the deviations from the mean for both the hours studied and the scores.
For the hours studied:
Deviation(hours) = (3 - 4.2), (5 - 4.2), (2 - 4.2), (8 - 4.2), (2 - 4.2), (4 - 4.2), (4 - 4.2), (5 - 4.2), (6 - 4.2), (3 - 4.2)
= -1.2, 0.8, -2.2, 3.8, -2.2, -0.2, -0.2, 0.8, 1.8, -1.2
For the scores:
Deviation(scores) = (65 - 76.3), (80 - 76.3), (60 - 76.3), (88 - 76.3), (66 - 76.3), (78 - 76.3), (85 - 76.3), (90 - 76.3), (80 - 76.3), (71 - 76.3)
= -11.3, 3.7, -16.3, 11.7, -10.3, 1.7, 8.7, 13.7, 3.7, -5.3
Step 3: Calculate the product of the deviations for each data point.
Product = Deviation(hours) x Deviation(scores)
= (-1.2)(-11.3), (0.8)(3.7), (-2.2)(-16.3), (3.8)(11.7), (-2.2)(-10.3), (-0.2)(1.7), (-0.2)(8.7), (0.8)(13.7), (1.8)(3.7), (-1.2)(-5.3)
= 13.56, 2.96, 35.86, 44.46, 22.66, -0.34, -1.74, 10.96, 6.66, 6.36
Step 4: Calculate the sum of the squared deviations for the hours studied.
Sum of Squares(hours) = Sum of (Deviation(hours))^2
= (-1.2)^2 + (0.8)^2 + (-2.2)^2 + (3.8)^2 + (-2.2)^2 + (-0.2)^2 + (-0.2)^2 + (0.8)^2 + (1.8)^2 + (-1.2)^2
= 16.44
Step 5: Calculate the sum of the products of the deviations.
Sum of Products = Sum of (Product)
= 13.56 + 2.96 + 35.86 + 44.46 + 22.66 - 0.34 - 1.74 + 10.96 + 6.66 + 6.36
= 142.52
Step 6: Calculate the slope of the regression line.
Slope = Sum of Products / Sum of Squares(hours)
= 142.52 / 16.44
≈ 8.660
Step 7: Calculate the y-intercept of the regression line.
y-intercept = Mean(scores) - (Slope * Mean(hours))
= 76.3 - (8.660 * 4.2)
≈ 57.506
Therefore, the equation of the regression line for the given data is:
y = 8.660x + 57.506
So, out of the options given, the correct answer is:
y = 8.660x + 57.506
To find the equation of the regression line for the given data, we need to calculate the slope and y-intercept first.
Step 1: Calculate the average of x-values (hours) and y-values (scores)
The average of x-values: (3 + 5 + 2 + 8 + 2 + 4 + 4 + 5 + 6 + 3) / 10 = 4.2
The average of y-values: (65 + 80 + 60 + 88 + 66 + 78 + 85 + 90 + 80 + 71) / 10 = 76.3
Step 2: Calculate the deviations of each x-value and y-value from their averages
Deviation of x-values: 3-4.2, 5-4.2, 2-4.2, 8-4.2, 2-4.2, 4-4.2, 4-4.2, 5-4.2, 6-4.2, 3-4.2
Deviation of y-values: 65-76.3, 80-76.3, 60-76.3, 88-76.3, 66-76.3, 78-76.3, 85-76.3, 90-76.3, 80-76.3, 71-76.3
Sum of the products of deviations: (3-4.2)(65-76.3) + (5-4.2)(80-76.3) + (2-4.2)(60-76.3) + (8-4.2)(88-76.3) + (2-4.2)(66-76.3) + (4-4.2)(78-76.3) + (4-4.2)(85-76.3) + (5-4.2)(90-76.3) + (6-4.2)(80-76.3) + (3-4.2)(71-76.3) = -18.98
Sum of the squares of x-value deviations: (3-4.2)^2 + (5-4.2)^2 + (2-4.2)^2 + (8-4.2)^2 + (2-4.2)^2 + (4-4.2)^2 + (4-4.2)^2 + (5-4.2)^2 + (6-4.2)^2 + (3-4.2)^2 = 21.6
Step 3: Calculate the slope (b) of the regression line
The slope (b) of the regression line is equal to the sum of the products of deviations divided by the sum of the squares of x-value deviations:
b = -18.98 / 21.6 = -0.8787
Step 4: Calculate the y-intercept (a) of the regression line
The y-intercept (a) of the regression line can be calculated using the formula:
a = average of y-values - (slope * average of x-values)
a = 76.3 - (-0.8787 * 4.2) = 76.3 + 3.682
So, the equation of the regression line for the given data is:
y = -0.8787x + 80.982
Therefore, the correct answer is:
y = -0.8787x + 80.982