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 39 59 60 65 66 Bone Density 338 316 313 312 311 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 = 289
Sum of Y = 1590
Mean X = 57.8
Mean Y = 318
Sum of squares (SS_X) = 478.8
Sum of products (SP) = -490

Regression Equation = ŷ = bX + a

b = SP/SS_X = -490/478.8 = -1.023

Step 2: a = M_Y - bM_X = 318 - (-1.02*57.8) = 377.152

Step 3: ŷ = -1.023X + 377.152

So for x=47, ŷ = (-1.023*45) + 377.152=331.117

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 -1.023

Step 5: Error prediction when x=47, is computed by

Step 6:

X Values
∑ = 289
Mean = 57.8
∑(X - M_x)² = SS_x = 478.8

Y Values
∑ = 1590
Mean = 318
∑(Y - M_y)² = SS_y = 514

X and Y Combined
N = 5
∑(X - M_x)(Y - M_y) = -490

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

r = -490 / √((478.8)(514)) = -0.988

So r^2=-0.988^2=0.976