The Coefficient of Determination

1
The Coefficient of Determination
The coefficient of determination, r2, is the
ratio of explained variation in y to the total
variation in y.
The correlation coefficient of number of times
absent and final grade is r 0.975. The
coefficient of determination is r2 (0.975)2
0.9506.
Interpretation About 95 of the variation in
final grades can be explained by the number of
times a student is absent. The other 5 is
unexplained and can be due to sampling error or
other variables such as intelligence, amount of
time studied, etc.
2
The Standard Error of Estimate
3
The Standard Error of Estimate
x
y
1 8 78 74.275 13.8756 2
2 92 97.819 33.8608 3 5
90 86.047 15.6262 4 12 58
58.579 0.3352 5 15 43 46.807
14.4932 6 9 74 70.351
13.3152 7 6 81 82.123 1.2611
92.767
Calculate
for each x.
4.307
4
Prediction Intervals
Given a specific linear regression equation and
x0, a specific value of x, a c-prediction
interval for y is
where
The point estimate is and E is the maximum
error of estimate.
Use a t-distribution with n 2 degrees of
freedom.
5
Application
Construct a 90 confidence interval for a final
grade when a student has been absent 6 times.
1. Find the point estimate
The point (6, 82.123) is the point on the
regression line with x-coordinate of 6.
6
Application
Construct a 90 confidence interval for a final
grade when a student has been absent 6 times.
2. Find E,
At the 90 level of confidence, the maximum error
of estimate is 9.438.
7
Application
Construct a 90 confidence interval for a final
grade when a student has been absent 6 times.
3. Find the endpoints.
E 82.123 9.438 72.685
E 82.123 9.438 91.561
72.685 lt y lt 91.561
When x 6, the 90 confidence interval is from
72.685 to 91.586.
8
Minitab Output
Regression Analysis The regression equation
is y 106 3.92x Predictor Coef
StDev T P Constant 105.668
3.655 28.91 0.000
x 3.9241 0.4019 9.76 0.000
S 4.307 R-Sq 95.0 R-Sq(adj) 94.0