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Statistics HW 5 NOTES: The Least-Squares Regression Line Part 3

Residuals A residual represents the difference between an actual observed value of the response variable and the value predicted by the regression line. Special property: the mean of the residuals is always zero.

residual = observed y – predicted y OR RESID = y y� � OR residual = A - P Residual plot x: explanatory variable (L1) y: residuals

(in L2: 2nd / STAT / 7:RESID / ENTER)

Note that you must have done the following before plotting the residuals. 9 Enter data into L1 & L2 9 Calculated the LSRL 9 Typed equation for LSRL it into “y=  ” 9 Turned Scatterplot to ON

Example 1: Let’s reexamine the data from the Kalama children. This data produces a favorable residual plot that indicates the line is a good model for the data. Age (months) 18 19 20 21 22 23 24 25 26 27 28 29 Height (cm) 76.1 77.0 78.1 78.2 78.8 79.7 79.9 81.1 81.2 81.8 82.8 83.5 LSRL: Interpret. Sketch the residual plot. Note: r = Example 2: Now try the following data set. This data produces a residual plot that indicates the line is not a good model.

LSRL: Interpret. Sketch the residual plot. Note: r = Example 3: Now try the following data set. This data also produces a residual plot that indicates the line begins to fail as a good model. xy

2 4 7 9 12 15 20 21 25 27 29 309 13 25 30 35 49 65 75 70 73 99 79

LSRL: Interpret. Sketch the residual plot. Note: r =

1 3 4 5 6 8 10 12 151 4.66 6.96 9.52 12.29 18.38 25.11 32.42 44.31

xy

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1. Success in hunting varies greatly among species of animals. Lions, who hunt singly, are rarely successful in more than 10 percent of their hunts. Wild African dogs, who hunt in packs, are among the most efficient of all hunters, succeeding at a rate of over 90 percent of their hunts.

In the early 1960’s, researcher Jane Goodall discovered that chimpanzees were not solely vegetarian in their diets, as had previously been thought. This discovery spurred a tremendous amount of primate research. Some of the latest primatology research has been done on chimpanzees to find out if larger hunting parties increase the chances of a successful hunt. The results of one such research project are summarized in the table for the number of chimpanzees in the hunting party versus the percentage of successful hunts.

Number of Chimps 1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 Percent of Success 20 30 28 42 40 58 45 62 65 63 75 75 78 75 82

a. Construct a scatter plot.

b. Determine the regression line.

c. Interpret the y-intercept. Does the interpretation make sense in this context?

d. Interpret the slope.

e. Find the correlation coefficient and interpret in terms of the problem.

f. Find the coefficient of determination and interpret in terms of the problem.

g. Sketch the residual plot. Interpret in terms of the problem.

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2. How quickly can athletes return to their sport following injuries requiring surgery? The paper “Arthroscopic Distal Clavicle Resection for Isolated Atraumatic Osteolysis in Weight Lifters” (American Journal of Sports Medicine, 1998) discovered their was a moderate positive (r = .55) linear relationship between a lifters age and the number of days after arthroscopic shoulder survgery before being able to return to their sport between 10 weight lifters. The average age of the weight lifters 30.4 with standard deviation of 2.875 years. The average number of days before being able to return to their sport was 3.2 days with a standard deviation of 1.398 days.

a. Determine the line to predict the number of days based on the age of the weight lifter.

b. Determine the coefficient of determination and interpret in terms of the problem.

c. Given the spread of the lifters was from 26 to 34 years old, predict the number of days for a 28 year old lifter. Do you feel this prediction is accurate? Explain.

3. The weights of children in the Egyptian village of Nahya were recorded. Here are the mean weights of the

170 children in that village: Age(months) 1 2 3 4 5 6 7 8 9 10 11 12 Weight(kg) 4.3 5.1 5.7 6.3 6.8 7.1 7.2 7.2 7.2 7.2 7.5 7.8

a. Make a scatterplot of mean weight against time.  Don’t forget to scale and label your axes appropriately.

b. Determine the equation of the LSRL for this data. Write the equation here and add to the scatterplot.

c. What is the correlation?

d. Sketch the residual plot. Interpret in terms of the problem.