Coefficient of Determination Section 3.2C
Coefficient of Determination
Section 3.2C
The regression line can be found using the calculator Put the data in L1 and L2.Press Stat – Calc - #8 (or 4) - enter
To get the correlation coefficient and coefficient of determination to show…Press 2nd catalog (0)Press DGo to Diagnostic on – press enter until you see
“done”
Using the Calculator
The following table lists the total weight lifted by the winners in eight weight classes of the 1996 Women’s National Weightlifting ChampionshipWeight Class (kg)
Total Lifted (kg)
46 14050 127.554 167.564 167.570 192.576 18583 200
1. Find LSRL
2. Find the correlation coefficient.
3. Find the residual for a 64 kg weight class.
4. Check out the residual plot.
If a line is appropriate, then we need to assess the accuracy of predictions based on the least squares line.
Coefficient of Determination
It’s the measure of the proportion of variability in the variable that can be “explained” by a linear relationship between the variables x and y.
Example# miles Cost
25 32.5
61 43.3
200 85
340 127
125 62.5
89 51.7
93 52.9
Rental Cost 25 0.3(Miles)
This relationship explains 100% of the variation in Cost.
But the line doesn’t always account for all of the variability.
Height Shoe Size
65 9
62 8.5
67 10
72 12
74 13
67 9.5
69 12
70 10
65 9
Shoe 16.03 .39 height
This doesn’t!
Total Sum of SquaresMeasures the total variation in the y-values.It’s the sum of squares of vertical distances
𝑺𝑺𝑻=∑ (𝒚 − 𝒚 )𝟐
Find the SST:
Height Shoe Size
65 9
62 8.5
67 10
72 12
74 13
67 9.5
69 12
70 10
65 9
Find the SST:
Height Shoe Size
65 9 1.7778
62 8.5 3.3611
67 10 .11111
72 12 2.7778
74 13 7.1111
67 9.5 .69444
69 12 2.7778
70 10 .11111
65 9 1.7778
𝑆𝑆𝑇=20.5
Sum of Squared Errors
This is the sum of the squared residuals
Total of the unexplained error
Formula:
Find the SSE:
Height Shoe Size
65 9 1.7778
62 8.5 3.3611
67 10 .11111
72 12 2.7778
74 13 7.1111
67 9.5 .69444
69 12 2.7778
70 10 .11111
65 9 1.7778
Find the SSE:
Height Shoe Size
65 9 1.7778 0.04478
62 8.5 3.3611 0.20543
67 10 .11111 1.4E-4
72 12 2.7778 0.00495
74 13 7.1111 0.08632
67 9.5 .69444 0.23833
69 12 2.7778 1.5258
70 10 .11111 1.3295
65 9 1.7778 0.04478
𝑆𝑆𝐸=3.48
Percent of unexplained error:
Coefficient of DeterminationIt’s the percent of variation in the y-
variable (response) that can be explained by the least-squares regression line of y on x.
Formula:
For height and shoe size – find and interpret the coefficient of determination.
𝑟2=1−𝑆𝑆𝐸𝑆𝑆𝑇
For height and shoe size – find and interpret the coefficient of determination.
Approximately 83% of the variation in shoe size can be explained by height.
Find the Coefficient of Determination:
Team Batting Avg.
Mean # runs per
game
0.289 5.9
0.279 5.5
0.277 4.9
0.274 5.2
0.271 4.9
0.271 5.4
0.268 4.5
0.268 4.6
0.266 5.1
Interpret this in context…
59.5% of the observed variability in mean number of runs per game can be explained by an approximate linear relationship between Team Batting average and mean runs per game.
Another example:
If r = 0.8, then what % can be explained by the least squares regression line?
Another example:A recent study discovered that the correlation between the age at which an infant first speaks and the child’s score on an IQ test given upon entering school is -0.68. A scatterplot of the data shows a linear form. Which of the following statements about this is true?A. Infants who speak at very early ages will have higher IQ scores by the beginning of elementary school than those who begin to speak later.B. 68% of the variation in IQ test scores is explained by the least- squares regression of age at first spoken word and IQ score.C. Encouraging infants to speak before they are ready can have a detrimental effect later in life, as evidenced by their lower IQ scores.D. There is a moderately strong, negative linear relationship between age at first spoken word and later IQ test score for the individuals this study.
HomeworkPage 192 (49, 51, 54, 56, 58, 71-78)