In: Statistics and Probability
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 runaverage, 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 |
Given:
Wins, x |
Earned runaverage, 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 runaverage, 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