Question

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.)

(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 60 kg. (Round your answer to three decimal places.)

g/cm²

Answer 1

Here as given is

a = 0.558

b = 0.009

n = 15

SST0 = 0.356

SSresid = 0.313

(a) Here we have to find the coefficient of determination

r2 = (SST0 - SSresid )/SST0 = (0.356 - 0.313)/0.356 = 0.1211 or 12.1%

(b) Here estimate of = sqrt [SSresid/dFresid]

dFresid = n -2 = 15 -2 = 13

estimate of = sqrt (0.313/13) = 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) Here b = 0.009

so here 1 kg weight increase in weight in age 13 will increase on average 0.009 g/cm2

(d) Here x = 60

y = 0.558 + 0.009 * 60= 1.098 g/cm2

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