Question

The table below gives the age and bone density for five randomly selected women. Using this data, consider the equation of the regression line, yˆ=b0+b1x for predicting a woman's bone density based on her age. 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.

Age	47	49	51	58	63
Bone Density	360	353	336	333	332

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=47 Round your answer to three decimal places.

Step 4 of 6: According to the estimated linear model, if the value of the independent variable is increased by one unit, then the change in the dependent variable yˆ is given by? (b0, b1, x, y)

Step 5 of 6: Find the error prediction when x=47. Round your answer to three decimal places.

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 = 268
Sum of Y = 1714
Mean X = 53.6
Mean Y = 342.8
Sum of squares (SS_X) = 179.2
Sum of products (SP) = -287.4

Regression Equation = ŷ = bX + a

b1 = SP/SS_X = -287.4/179.2 = -1.604

Step 2: b0 = M_Y - bM_X = 342.8 - (-1.6*53.6) = 428.763

ŷ = -1.604X + 428.763

Step 3: For x=47, ŷ = (-1.604*47) + 428.763=353.375

Step 4:  If the value of the independent variable is increased by one unit, then the change in the dependent variable yˆ is given by b1=-1.604

Step 5: Error prediction is

Step 6:

X Values
∑ = 268
Mean = 53.6
∑(X - M_x)² = SS_x = 179.2

Y Values
∑ = 1714
Mean = 342.8
∑(Y - M_y)² = SS_y = 658.8

X and Y Combined
N = 5
∑(X - M_x)(Y - M_y) = -287.4

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

r = -287.4 / √((179.2)(658.8)) = -0.837

So Coefficient of determination is