Question

#3

The authors of the paper "Weight-Bearing Activity during Youth Is a More Important Factor for Peak Bone Mass than Calcium Intake" studied a number of variables they thought might be related to bone mineral density (BMD). The accompanying data on

x = weight

at age 13 and

y = bone

mineral density at age 27 are consistent with summary quantities for women given in the paper.

Weight (kg)	BMD (g/cm2)
54.4	1.15
59.3	1.26
74.6	1.42
62.0	1.06
73.7	1.44
70.8	1.02
66.8	1.26
66.7	1.35
64.7	1.02
71.8	0.91
69.7	1.28
64.7	1.17
62.1	1.12
68.5	1.24
58.3	1.00

A simple linear regression model was used to describe the relationship between weight at age 13 and BMD at age 27. The following values are given for this data.

a = 0.558           b = 0.009         n = 15
SSTo = 0.356     SSResid = 0.313

(a)

What percentage of observed variation in BMD at age 27 can be explained by the simple linear regression model? (Round your answer to one decimal place.)

%

(b)

Give an estimate of σ. (Round your answer to three decimal places.)

Interpret this estimate.

This is a typical deviation of a weight value in the sample from the value predicted by the least-squares line.This is a typical deviation of a bone mineral density value in the sample from the value predicted by the least-squares line.    This is an average of bone mineral density values in the sample from the value predicted by the least-squares line.This is an average of weight values in the sample from the value predicted by the least-squares line.

(c)

Give an estimate of the average change in BMD associated with a 1 kg increase in weight at age 13. (Round your answer to three decimal places.)

g/cm²

(d)

Compute an estimate of the mean BMD at age 27 for women whose weight at age 13 was 55 kg. (Round your answer to three decimal places.)

g/cm²

Answer 1

(a)

SSRegression = SSTo - SSResid = 0.356 - 0.313 = 0.043

R-squared = SSRegression / SSTo = 0.043 / 0.356 = 0.121

Percentage of observed variation in BMD at age 27 can be explained by the simple linear regression model is

12.1%

(b)

σ = = = 0.155

This is a typical deviation of a bone mineral density value in the sample from the value predicted by the least-squares line.

(c)

The slope value is b = 0.009

The estimate of the average change in BMD associated with a 1 kg increase in weight at age 13 is 0.009 g/cm²

(d)

The regression equation is,

y = 0.558 + 0.009x

For x = 55 kg,

y = 0.558 + 0.009 * 55 = 1.053 g/cm²