Question

The table below gives the number of hours spent unsupervised each day as well as the overall grade averages for seven randomly selected middle school students. Using this data, consider the equation of the regression line, yˆ=b0+b1x, for predicting the overall grade average for a middle school student based on the number of hours spent unsupervised each day. 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.

Hours Unsupervised	0	0.5	1	1.5	2	3.5	4
Overall Grades	89	81	73	72	69	67	63

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=0.5. Round your answer to three decimal places.

Step 4 of 6: Determine if the statement "All points predicted by the linear model fall on the same line" is true or false.

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 = 12.5
Sum of Y = 514
Mean X = 1.7857
Mean Y = 73.4286
Sum of squares (SS_X) = 13.4286
Sum of products (SP) = -71.8571

Regression Equation = ŷ = bX + a

b1 = SP/SS_X = -71.86/13.43 = -5.351

Step 2: b0 = M_Y - b1M_X = 73.43 - (-5.35*1.79) = 82.984

ŷ = -5.351X + 82.984

Step 3: For x=0.5,

ŷ = (-5.351*0.5) + 82.984=80.533

Step 4:

So answer here is False

Step 5: If the value of the independent variable is increased by one unit, then find the change in the dependent variable yˆ is slope and value of slope is b1=-5.351

Step 6:

X Values
∑ = 12.5
Mean = 1.786
∑(X - M_x)² = SS_x = 13.429

Y Values
∑ = 514
Mean = 73.429
∑(Y - M_y)² = SS_y = 471.714

X and Y Combined
N = 7
∑(X - M_x)(Y - M_y) = -71.857

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

r = -71.857 / √((13.429)(471.714)) = -0.903

So coefficient of determination is R^2=-0.903^2=0.815