Question

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

Answer 1

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