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Math  /  Data & Statistics

QuestionIn a multiple linear regression of E(Y∣X1=x1,X2=x2)=a+b1x1+b2x2E\left(Y \mid X_{1}=x_{1}, X_{2}=x_{2}\right)=a+b_{1} x_{1}+b_{2} x_{2}, we get a table of coefficients like this:
Coefficient Value a 2.02.0 b1b_{1} -1 b2b_{2} 0.5
Consider two groups that have the following characteristics: - The same values of X2X_{2} - Group A has X1=2X_{1}=2 - Group B has X1=1X_{1}=1
Let E(Y∣A)E(Y \mid A) be the average of YY within group A, and E(Y∣B)E(Y \mid B) be the average of YY within group B.
According to this model, which group has the larger conditional mean of YY ? Group A Group B They are the same. We cannot answer without knowing the value of X2X_{2}.

Studdy Solution

STEP 1

1. The regression equation is given by E(Y∣X1=x1,X2=x2)=a+b1x1+b2x2 E(Y \mid X_1 = x_1, X_2 = x_2) = a + b_1 x_1 + b_2 x_2 .
2. The coefficients are a=2.0 a = 2.0 , b1=−1 b_1 = -1 , and b2=0.5 b_2 = 0.5 .
3. Group A has X1=2 X_1 = 2 and Group B has X1=1 X_1 = 1 .
4. Both groups have the same value of X2 X_2 .

STEP 2

1. Calculate E(Y∣A) E(Y \mid A) using the regression equation.
2. Calculate E(Y∣B) E(Y \mid B) using the regression equation.
3. Compare E(Y∣A) E(Y \mid A) and E(Y∣B) E(Y \mid B) .

STEP 3

Substitute X1=2 X_1 = 2 into the regression equation for Group A:
E(Y∣A)=a+b1⋅2+b2⋅X2 E(Y \mid A) = a + b_1 \cdot 2 + b_2 \cdot X_2
E(Y∣A)=2.0−1⋅2+0.5⋅X2 E(Y \mid A) = 2.0 - 1 \cdot 2 + 0.5 \cdot X_2
E(Y∣A)=2.0−2+0.5⋅X2 E(Y \mid A) = 2.0 - 2 + 0.5 \cdot X_2
E(Y∣A)=0+0.5⋅X2 E(Y \mid A) = 0 + 0.5 \cdot X_2

STEP 4

Substitute X1=1 X_1 = 1 into the regression equation for Group B:
E(Y∣B)=a+b1⋅1+b2⋅X2 E(Y \mid B) = a + b_1 \cdot 1 + b_2 \cdot X_2
E(Y∣B)=2.0−1⋅1+0.5⋅X2 E(Y \mid B) = 2.0 - 1 \cdot 1 + 0.5 \cdot X_2
E(Y∣B)=2.0−1+0.5⋅X2 E(Y \mid B) = 2.0 - 1 + 0.5 \cdot X_2
E(Y∣B)=1.0+0.5⋅X2 E(Y \mid B) = 1.0 + 0.5 \cdot X_2

STEP 5

Compare E(Y∣A) E(Y \mid A) and E(Y∣B) E(Y \mid B) :
E(Y∣A)=0+0.5⋅X2 E(Y \mid A) = 0 + 0.5 \cdot X_2
E(Y∣B)=1.0+0.5⋅X2 E(Y \mid B) = 1.0 + 0.5 \cdot X_2
Since E(Y∣B)=1.0+E(Y∣A) E(Y \mid B) = 1.0 + E(Y \mid A) , Group B has the larger conditional mean of Y Y .
The group with the larger conditional mean of Y Y is:
Group B \text{Group B}

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