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 35 41 52 56 66 Bone Density 358 350 348 332 321 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 if the statement "Not all points predicted by the linear model fall on the same line" is true or false. Step 4 of 6: Determine the value of the dependent variable ˆy at x = 0. Step 5 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 ˆy is given by? 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 = 250
Sum of Y = 1709
Mean X = 50
Mean Y = 341.8
Sum of squares (SS_X) = 602
Sum of products (SP) = -696

Regression Equation = ŷ = bX + a

b = SP/SS_X = -696/602 = -1.156

Step 2: a = M_Y - bM_X = 341.8 - (-1.16*50) = 399.607

Step 3:

So answer is true

Step 4: Regression equation is ŷ = -1.156X + 399.607

So for x=0, y=399.607

Step 5: This is slope value and as per equation it is -1.156

Step 6:

X Values
∑ = 250
Mean = 50
∑(X - M_x)² = SS_x = 602

Y Values
∑ = 1709
Mean = 341.8
∑(Y - M_y)² = SS_y = 896.8

X and Y Combined
N = 5
∑(X - M_x)(Y - M_y) = -696

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

r = -696 / √((602)(896.8)) = -0.947

So r^2=0.897