Question

In: Statistics and Probability

The accompanying data are the number of wins and the earned run averages​ (mean number of...

The accompanying data are the number of wins and the earned run averages​ (mean number of earned runs allowed per nine innings​ pitched) for eight baseball pitchers in a recent season. Find the equation of the regression line. Then construct a scatter plot of the data and draw the regression line. Then use the regression equation to predict the value of y for each of the given​ x-values, if meaningful. If the​ x-value is not meaningful to predict the value of​ y, explain why not.

​(a) x=5 wins ​(b) x=10 wins ​(c) x=21 wins ​(d) x=15 wins. The equation of the regression line is y with equals ____xplus ______. ​(Round to two decimal places as​ needed.)

​Wins, x

Earned run​average, y

20

2.75

18

3.32

17

2.66

16

3.77

14

3.88

12

4.25

11

3.76

9

5.06

Solutions

Expert Solution

Given:

Wins, x

Earned run​average, y

20

2.75

18

3.32

17

2.66

16

3.77

14

3.88

12

4.25

11

3.76

9

5.06

In general, the linear regression equation is given as

Y = b0 + b1 X

b0 and b1 are obtained solving the following normal equations

n = number of observations = 8

Wins, X

Earned run​average, Y

XY

X2

20

2.75

55

400

18

3.32

59.76

324

17

2.66

45.22

289

16

3.77

60.32

256

14

3.88

54.32

196

12

4.25

51

144

11

3.76

41.36

121

9

5.06

45.54

81

The above values are plugged in to the above equations

Multiplying the equation (1) by 117 and multiplying the equation (2) by 8 we will have

Subtracting (4) from (3), we will have

Substituting b1 = -0.1821 in equation (1) we will have

Now, we have b0 = 6.34 and b1 = -0.18

Therefore, the regression equation is given as

y = - 0.18 x + 6.34

The scatter plot for the given data is given as

To plot, the regression line on the plot, the predicted value for each of the given x values are calculated using the regression equation as follows

Wins, X

Earned run ​average, Y

Predicted Y

20

2.75

-0.18 (20) + 6.34 = 2.74

18

3.32

-0.18 (18) + 6.34 = 3.10

17

2.66

-0.18 (17) + 6.34 = 3.28

16

3.77

-0.18 (16) + 6.34 = 3.46

14

3.88

-0.18 (14) + 6.34 = 3.82

12

4.25

-0.18 (12) + 6.34 = 4.18

11

3.76

-0.18 (11) + 6.34 = 4.36

9

5.06

-0.18 (9) + 6.34 = 4.72

The above predicted values are plotted in the scatter plot

The predicted values for the given x values are calculated as below

​(a) x=5 wins

Y = -0.18 (5) + 6.34 = 5.44

​(b) x=10 wins

Y = -0.18 (10) + 6.34 = 4.54

​(c) x=21 wins

Y = -0.18 (21) + 6.34 = 2.56

​(d) x=15 wins.

Y = -0.18 (15) + 6.34 = 3.64


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