Questions: Calculate the predicted Y(Ŷ ) for each X value (put in table below). ( 3 pts ) X Y 2 0 5 1 6 4 6 5 3 6 MX=4 MY=3

Solution Steps

To calculate the predicted \( \hat{Y} \) for each \( X \) value, we need to use the linear regression formula. First, calculate the slope (\( b \)) using the formula \( b = \frac{\sum{(X_i - M_X)(Y_i - M_Y)}}{\sum{(X_i - M_X)^2}} \). Then, calculate the intercept (\( a \)) using \( a = M_Y - b \times M_X \). Finally, use the equation \( \hat{Y} = a + b \times X \) to find the predicted \( \hat{Y} \) for each \( X \).

To find the slope \( b \) of the regression line, use the formula:

\[ b = \frac{\sum{(X_i - M_X)(Y_i - M_Y)}}{\sum{(X_i - M_X)^2}} \]

Given:

• \( \sum{(X_i - M_X)(Y_i - M_Y)} = 7 \)
• \( \sum{(X_i - M_X)^2} = 14 \)

Thus, the slope is:

\[ b = \frac{7}{14} = \frac{1}{2} \]

The intercept \( a \) is calculated using the formula:

\[ a = M_Y - b \times M_X \]

Substituting the known values:

\[ a = 3 - \frac{1}{2} \times 4 = 3 - 2 = 1 \]

Using the linear equation \( \hat{Y} = a + b \times X \), calculate the predicted \( \hat{Y} \) for each \( X \):

• For \( X = 2 \): \[ \hat{Y} = 1 + \frac{1}{2} \times 2 = 1 + 1 = 2 \]

• For \( X = 5 \): \[ \hat{Y} = 1 + \frac{1}{2} \times 5 = 1 + 2.5 = 3.5 \]

• For \( X = 6 \): \[ \hat{Y} = 1 + \frac{1}{2} \times 6 = 1 + 3 = 4 \]

• For \( X = 6 \) (again): \[ \hat{Y} = 1 + \frac{1}{2} \times 6 = 1 + 3 = 4 \]

• For \( X = 3 \): \[ \hat{Y} = 1 + \frac{1}{2} \times 3 = 1 + 1.5 = 2.5 \]

Final Answer