Question

The table below gives the number of parking tickets received in one semester and the GPA for five randomly selected college students who drive to campus. Using this data, consider the equation of the regression line, yˆ=b0+b1x , for predicting the GPA of a college student who drives to campus based on the number of parking tickets they receive in one semester. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant. Number of Tickets 1 2 3 7 8 GPA 3.7 3.3 2.9 2 1.5 Table

Step 1 of 6: Find the estimated slope. Round your answer to three decimal places.

Step 2 of 6: Find the estimated y-intercept. Round your answer to three decimal places.

Step 3 of 6: Find the estimated value of y when x = 34. Round your answer to three decimal places.

Step 4 of 6: Determine the value of the dependent variable yˆ at x = 0.

Step 5 of 6: Substitute the values you found in steps 1 and 2 into the equation for the regression line to find the estimated linear model. According to this model, if the value of the independent variable is increased by one unit, then find the change in the dependent variable yˆ.

Step 6 of 6: Find the value of the coefficient of determination. Round your answer to three decimal places.

Answer 1

Step 1.

Sum of X = 21
Sum of Y = 13.4
Mean X = 4.2
Mean Y = 2.68
Sum of squares (SS_X) = 38.8
Sum of products (SP) = -11.28

Regression Equation = ŷ = bX + a

b = SP/SS_X = -11.28/38.8 = -0.291

Step 2: a = M_Y - bM_X = 2.68 - (-0.29*4.2) = 3.901

Step 3: ŷ = -0.291X + 3.901

For x=34, ŷ = (-0.291*34) + 3.901=-5.993

Or x=3.4,  ŷ = (-0.291*3.4) + 3.901=2.912

Step 4: For x=0, y=3.901

Step 5: ŷ = -0.291X + 3.901

If the value of the independent variable is increased by one unit, then the change in the dependent variable yˆ is slope and its value is -0.291

Step 6:

X Values
∑ = 21
Mean = 4.2
∑(X - M_x)² = SS_x = 38.8

Y Values
∑ = 13.4
Mean = 2.68
∑(Y - M_y)² = SS_y = 3.328

X and Y Combined
N = 5
∑(X - M_x)(Y - M_y) = -11.28

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = -11.28 / √((38.8)(3.328)) = -0.993

So r^2=-0.993^2=0.986

Hence 98.6% of variation in y is explained by x