Question

The table below gives the number of hours ten randomly selected students spent studying and their corresponding midterm exam grades. Using this data, consider the equation of the regression line, yˆ=b0+b1x, for predicting the midterm exam grade that a student will earn based on the number of hours spent studying. 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 Studying	0	0.5	1.5	2	2.5	3	3.5	4	4.5	5.5
Midterm Grades	63	68	72	73	80	84	86	87	94	99

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:

Determine the value of the dependent variable yˆ at x=0. (b0, b1, x or y)

Step 4 of 6:

Find the estimated value of y when x=2.5. Round your answer to three decimal places.

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.

(X-MJ2 Y- M 7.29 4.84 1.44 0.49 0.04 0.09 0.64 1.69 3.24 7.84 47.52 27.72 10.32 5.32 0.12 1.02 4.32 8.32 24.12 51.52 -2.7 -2.2 -12.6 8.6 7.6 0.6 3.4 5.4 6.4 13.4 18.4 -0.7 0.2 0.3 0.8 1.3 1.8 2.8 SS: 27.6 SP: 180.3

Sum of X = 27
Sum of Y = 806
Mean X = 2.7
Mean Y = 80.6
Sum of squares (SS_X) = 27.6
Sum of products (SP) = 180.3

Regression Equation = ŷ = bX + a

b = SP/SS_X = 180.3/27.6 = 6.533

Step 2. a = M_Y - bM_X = 80.6 - (6.53*2.7) = 62.962

Step 3: Regression line is

ŷ = 6.533X + 62.962

For x=0, y=62.962

Step 4: For x=2.5, ŷ = (6.533*2.5)+ 62.962=79.295

Step 5: Here we need to find value of y when x changes to 1 and that is value of slope=6.533

Step 6:

(X-MJ2 Y- M 7.29 4.84 1.44 0.49 0.04 0.09 0.64 1.69 3.24 7.84 47.52 27.72 10.32 5.32 0.12 1.02 4.32 8.32 24.12 51.52 -2.7 -2.2 -12.6 8.6 7.6 0.6 3.4 5.4 6.4 13.4 18.4 -0.7 0.2 0.3 0.8 1.3 1.8 2.8 SS: 27.6 SP: 180.3

X Values
∑ = 27
Mean = 2.7
∑(X - M_x)² = SS_x = 27.6

Y Values
∑ = 806
Mean = 80.6
∑(Y - M_y)² = SS_y = 1200.4

X and Y Combined
N = 10
∑(X - M_x)(Y - M_y) = 180.3

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

r = 180.3 / √((27.6)(1200.4)) = 0.991

So r^2=0.991^2=0.982