Question

Metabolic rate, the rate at which the body consumes energy, is important in studies of weight gain, dieting, and exercise. The provided table gives data on the lean body mass and resting metabolic rate for 12 women and 7 men who are subjects in a study of dieting. Lean body mass, given in kilograms, is a person’s weight leaving out all fat. Metabolic rate is measured in kilocalories (Cal) burned per 24 hours, the same calories used to describe the energy content of foods. The researchers believe that lean body mass is an important influence on metabolic rate.

Subject	Sex	Mass	Rate
1	M	62.0	1792
2	M	62.9	1666
3	F	36.1	995
4	F	54.6	1425
5	F	48.5	1396
6	F	42.0	1418
7	M	47.4	1362
8	F	50.6	1502
9	F	42.0	1256
10	M	48.7	1614
11	F	40.3	1189
12	F	33.1	913
13	M	51.9	1460
14	F	42.4	1124
15	F	34.5	1052
16	F	51.1	1347
17	F	41.2	1204
18	M	51.9	1867
19	M	46.9	14.39

The parameter associated with the interaction term is often used to decide if a model with parallel regression lines can be used. Test the hypothesis that this parameter is equal to zero, and comment on whether you would be willing to use the more restrictive model with parallel regression lines for these data. Find the t statistic. (Enter your answer rounded to two decimal places.)

t=

Find the P‑value. (Enter your answer rounded to three decimal places.)

P=

Find R². (Enter your answer rounded to one decimal places.)

R²=

Answer 1

Let's run the regression analysis in R or any other software of your choice. The code is:

sex<-c(1,1,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,1,1)

Mass<-c(62 ,62.9
         ,36.1
         ,54.6
         ,48.5
         ,42
         ,47.4
         ,50.6
         ,42
         ,48.7
         ,40.3
         ,33.1
         ,51.9
         ,42.4
         ,34.5
         ,51.1
         ,41.2
         ,51.9
         ,46.9
 )

Rate<-c(1792
         ,1666
         ,995
         ,1425
         ,1396
         ,1418
         ,1362
         ,1502
         ,1256
         ,1614
         ,1189
         ,913
         ,1460
         ,1124
         ,1052
         ,1347
         ,1204
         ,1867
         ,1439
 )

Model1<-lm(Rate~Sex + Mass + Sex*Mass)

summary(Model1)

The results are:

Call:
lm(formula = Rate ~ sex + Mass + sex * Mass)

Residuals:
    Min      1Q  Median 
-142.52  -85.70   12.96 
     3Q     Max 
  44.33  287.10 

Coefficients:
            Estimate
(Intercept)  201.162
sex          509.341
Mass          24.026
sex:Mass      -7.275
            Std. Error t value
(Intercept)    236.613   0.850
sex            468.200   1.088
Mass             5.435   4.420
sex:Mass         9.309  -0.781
            Pr(>|t|)    
(Intercept) 0.408595    
sex         0.293823    
Mass        0.000496 ***
sex:Mass    0.446679    
---
Signif. codes:  
  0 ‘***’ 0.001 ‘**’ 0.01
  ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 123.8 on 15 degrees of freedom
Multiple R-squared:  0.8073,   Adjusted R-squared:  0.7688 
F-statistic: 20.95 on 3 and 15 DF,  p-value: 1.277e-05

Let's check whether the coefficient of the interaction term is zero.

Ho: B3 = 0

Ha: B3 =/ 0

The parameter associated with the interaction term, Sex:Mass is highlighted in Bold. The t-statistic = -0.78

p-value = 0.447

As the p-value > 0.05, we fail to reject Ho and we conclude that we would be willing to use the more restrictive model with parallel regression lines for these data.

R-square = 0.81