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 Size65 962 8.567 1072 1274 1367 9.569 1270 1065 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 Size65 962 8.567 1072 1274 1367 9.569 1270 1065 9

Find the SST:

Height Shoe Size65 9 1.777862 8.5 3.361167 10 .1111172 12 2.777874 13 7.111167 9.5 .6944469 12 2.777870 10 .1111165 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 Size65 9 1.777862 8.5 3.361167 10 .1111172 12 2.777874 13 7.111167 9.5 .6944469 12 2.777870 10 .1111165 9 1.7778

Find the SSE:

Height Shoe Size65 9 1.7778 0.0447862 8.5 3.3611 0.2054367 10 .11111 1.4E-472 12 2.7778 0.0049574 13 7.1111 0.0863267 9.5 .69444 0.2383369 12 2.7778 1.525870 10 .11111 1.329565 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)