Question

Three percent of a man's body is essential fat; for a woman the percentage is closer to 12.5%. As the name implies, essential fat is necessary for a normal healthy body. Fat is stored in small amounts throughout the body. Too much fat, however, can be dangerous to your health. For men between 18 and 39 years old, a healthy percent of body fat ranges from 8% to 19%; for women of the same age, it's 21% to 32%.

Measuring body fat can be tedious and expensive. The "standard reference" measurement is by dual-eneregy X-ray absorptiometry (DEXA), which involves two low-dose X-ray generators and takes from 10 to 20 minutes.

Because of the time and expense involved with the DEXA method, health professionals would like to be able to make a useable prediction of body fat from easily measurable variables such as weight or waist size.This Excel file shows the waist size (inches), weight (pounds) and percent body fat for 20 individuals.

Question 1. What is the slope of the least squares regression line of %body fat on waist size?

  (use 4 decimal places).

Question 2. Find s_b1, the estimate of the standard deviation σ_b1 of the least squares slope b₁ (use 4 decimal places).

  s_b1, estimate of σ_b1

Question 3. Determine the 99% confidence interval for the slope of the least squares regression line of %body fat on waist size.

lower bound
upper bound

Question 4. Determine a 99% confidence interval for the mean %body fat found in people with 40-inch waists (use 2 decimal places).

lower bound
upper bound

Question 5. Determine a 99% prediction interval for the %body fat of an individual with a 40-inch waist (use 2 decimal places).

lower bound
upper bound

Answer 1

Question 1. What is the slope of the least squares regression line of %body fat on waist size?

  (use 4 decimal places).

2.2215

Question 2. Find sb1, the estimate of the standard deviation σb1 of the least squares slope b1 (use 4 decimal places).

  sb1 = 0.2728

estimate of σ = 4.5397

b1 = 2.2215

Question 3. Determine the 99% confidence interval for the slope of the least squares regression line of %body fat on waist size.

lower bound = 1.44
upper bound = 3.01

Question 4. Determine a 99% confidence interval for the mean %body fat found in people with 40-inch waists (use 2 decimal places).

lower bound = 22.57
upper bound = 30.03

Question 5. Determine a 99% prediction interval for the %body fat of an individual with a 40-inch waist (use 2 decimal places).

lower bound = 12.71
upper bound = 39.89

Waist (in.)	Weight (lb)	Body Fat (%)
32	175	6
36	181	21
38	200	15
33	159	6
39	196	22
40	192	31
41	205	32
35	173	21
38	187	25
38	188	30
33	188	10
40	240	20
36	175	22
32	168	9
44	246	38
33	160	10
41	215	27
34	159	12
34	146	10
44	219	28
	r²	0.787
	r	0.887
	Std. Error	4.5397
	n	20
	k	1
	Dep. Var.	Body Fat (%)
ANOVA table
Source	SS	df	MS	F	p-value
Regression	1,366.790	1	1,366.790	66.32	1.90E-07
Residual	370.960	18	20.6089
Total	1,737.750	19
Regression output				confidence interval
variables	coefficients	std. error	t (df=18)	p-value	99% lower	99% upper
Intercept	-62.5573
Waist (in.)	2.2215	0.2728	8.144	1.90E-07	1.44	3.01
Predicted values for: Body Fat (%)
		99% Confidence Interval	99% Prediction Interval
Waist (in.)	Predicted	lower	upper	lower	upper	Leverage
40	26.303	22.57	30.03	12.71	39.89	0.081