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	38	40	46	68	69
Bone Density	352	335	328	327	322

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: Find the estimated value of y when x=46. 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 of 6:

From the given data, the following Table is calculated:

X	Y	XY	X2	Y2
38	352	13376	1444	123904
40	335	13400	1600	112225
46	328	15088	2116	107584
68	327	22236	4624	106929
69	322	22218	4761	103684
Total = 261	1664	86318	14545	554326

The estimated slope is given by:

Step 2 of 6:

Estimated y - intercept is given by:

Step 3 of 6:

Correct option:

True

Explanation:

The linear model is given by:

For x = 38, we get:

The predicted value of 341.189 is different from actual value of 352.

Step 4 of 6:

For x = 46, we get:

Step 5 of 6:

Substituting the values you found in steps 1 and 2 into the equation for the regression line, we find the estimated linear model as follows:

According to this model, if the value of the independent variable is increased by one unit, then the change in the dependent variable yˆ = - 0.589

Step 6 of 6:

Correlation Coefficient (r) is given by:

So,

the value of the coefficient of determination (R²) is given by:

R² = (-0.765)² = 0.585