You are given the following data.
# of Absences Final Grade
0 96
1 92
2 71
3 66
4 60
5 51
A. Find the correlation coefficient for the data.
B. Find the equation for the regression line for the data, and predict the final grade of a student who misses 3.5 days.
To find the correlation coefficient and the equation for the regression line, we need to calculate several values: the mean of the absences, the mean of the final grades, the sum of the products of the absences and the grades, the sum of the squared absences, and the sum of the squared grades.
Here are the steps to calculate the correlation coefficient and the equation for the regression line:
1. Calculate the mean of the absences ("x̄") and the mean of the final grades ("ȳ"):
x̄ = (0 + 1 + 2 + 3 + 4 + 5) / 6 = 15 / 6 = 2.5 absences
ȳ = (96 + 92 + 71 + 66 + 60 + 51) / 6 = 436 / 6 = 72.67
2. Calculate the sum of the products of the absences and the grades ("∑xy"), the sum of the squared absences ("∑x²"), and the sum of the squared grades ("∑y²"):
∑xy = (0 * 96) + (1 * 92) + (2 * 71) + (3 * 66) + (4 * 60) + (5 * 51) = 0 + 92 + 142 + 198 + 240 + 255 = 927
∑x² = (0²) + (1²) + (2²) + (3²) + (4²) + (5²) = 0 + 1 + 4 + 9 + 16 + 25 = 55
∑y² = (96²) + (92²) + (71²) + (66²) + (60²) + (51²) = 9216 + 8464 + 5041 + 4356 + 3600 + 2601 = 33678
3. We can use the following formulas to calculate the correlation coefficient ("r") and the regression line equation:
r = (n * ∑xy - ∑x * ∑y) / sqrt((n * ∑x² - (∑x)²) * (n * ∑y² - (∑y)²))
regression line equation:
y = a + bx
where a = (ȳ - b * x̄)
b = [(n * ∑xy) - (∑x * ∑y)] / [(n * ∑x²) - (∑x)²]
4. Substitute the values into the formulas:
r = (6 * 927 - (15 * 436)) / sqrt((6 * 55 - (15)²) * (6 * 33678 - (436)²))
b = [(6 * 927) - (15 * 436)] / [(6 * 55) - (15)²]
a = (72.67 - b * 2.5)
5. Calculate the correlation coefficient ("r") and the regression line equation:
r = 0.929
b ≈ -12.89
a ≈ 105.33
Now that we have the correlation coefficient and the regression line equation, we can predict the final grade of a student who misses 3.5 days:
Predicted Final Grade = a + b * x
= 105.33 + (-12.89) * 3.5
= 105.33 - 45.11
= 60.22
Therefore, the predicted final grade for a student who misses 3.5 days is approximately 60.22.