Question

3.

3. 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	41	44	45	60	62
Bone Density	355	353	345	336	315

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: 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 5 of 6: Find the estimated value of y when x=44. 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

Solution:

Here, we have to construct the regression model for the prediction of the dependent variable bone density based on the independent variable age. Required regression model is given as below:

Regression Statistics
Multiple R	0.908424493
R Square	0.82523506
Adjusted R Square	0.76698008
Standard Error	7.846264266
Observations	5
ANOVA
	df	SS	MS	F	Significance F
Regression	1	872.1084112	872.1084112	14.16591438	0.032805415
Residual	3	184.6915888	61.56386293
Total	4	1056.8
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	416.635514	20.45211603	20.37126689	0.00025862	351.5477529	481.7232751
Age	-1.504672897	0.399778829	-3.763763327	0.032805415	-2.776947556	-0.232398239

Step 1

The estimated slope for this regression model is given as -1.505.

Step 2

Estimated y-intercept for this regression model is given as 416.636.

Step 3

Given statement is false because we know that the predicted values of y always falls on the same regression line.

Step 4

Required regression equation is given as below:

Y = 416.636 – 1.505*X

According to this model, if the value of the independent variable is increased by one unit, the the dependent variable yˆ is decreased by 1.505.

Step 5

We are given x = 44

Y = 416.636 – 1.505*X

Y = 416.636 - 1.505*44

Y = 350.416

Step 6

The coefficient of determination or the value of R square is given as 0.825, which means about 82.5% of the variation in the dependent variable bone density is explained by the independent variable age.