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 for the given data, you can use the following steps:
Step 1: Calculate the mean (average) of both sets of data.
Mean of # of Absences (x̄) = (0 + 1 + 2 + 3 + 4 + 5)/6 = 15/6 ≈ 2.5
Mean of Final Grade (ȳ) = (96 + 92 + 71 + 66 + 60 + 51)/6 = 436/6 ≈ 72.67
Step 2: Calculate the sum of the products of each pair of values.
Sum of Products (Σxy) = (0 * 96) + (1 * 92) + (2 * 71) + (3 * 66) + (4 * 60) + (5 * 51) = 0 + 92 + 142 + 198 + 240 + 255 = 927
Step 3: Calculate the sum of the squares for each set of data.
Sum of Squares for # of Absences (Σx^2) = (0^2) + (1^2) + (2^2) + (3^2) + (4^2) + (5^2) = 0 + 1 + 4 + 9 + 16 + 25 = 55
Sum of Squares for Final Grade (Σy^2) = (96^2) + (92^2) + (71^2) + (66^2) + (60^2) + (51^2) = 9216 + 8464 + 5041 + 4356 + 3600 + 2601 = 33478
Step 4: Calculate the standard deviations for both sets of data.
Standard Deviation for # of Absences (σx) = sqrt((Σx^2)/n - (x̄^2)) = sqrt((55/6) - (2.5^2)) = sqrt((55/6) - 6.25) ≈ 1.87
Standard Deviation for Final Grade (σy) = sqrt((Σy^2)/n - (ȳ^2)) = sqrt((33478/6) - (72.67^2)) = sqrt((5579.67) - (5282.69)) ≈ 11.69
Step 5: Calculate the correlation coefficient (r).
r = (Σxy - (n * x̄ * ȳ))/(σx * σy) = (927 - (6 * 2.5 * 72.67))/(1.87 * 11.69) = (927 - 436.02)/(21.88) ≈ 23.98
Therefore, the correlation coefficient for the given data is approximately 23.98.
To find the equation for the regression line and predict the final grade of a student who misses 3.5 days, you can use the following steps:
Step 1: Calculate the slope (b) of the regression line.
b = (r * σy)/σx = (23.98 * 11.69)/1.87 ≈ 150.75
Step 2: Calculate the y-intercept (a) of the regression line.
a = ȳ - (b * x̄) = 72.67 - (150.75 * 2.5) = 72.67 - 376.88 = -304.21
Step 3: Write the equation of the regression line in slope-intercept form (y = mx + b).
Regression Line Equation: y = 150.75x - 304.21
Step 4: Predict the final grade of a student who misses 3.5 days using the regression line equation.
Predicted Final Grade (y) = 150.75 * 3.5 - 304.21 ≈ 56.79
Therefore, the predicted final grade of a student who misses 3.5 days is approximately 56.79.