Stat 112 Practice Quiz 1
1. It is extremely difficult to measure the volume of a child because of a child’s irregular shape but it is easy to measure the weight of a child. Regression analysis can be used to estimate a child’s volume based on a child’s weight. A study was done in which the volume, measured in cubic decimeters, of 18 children was determined by an elaborate procedure along with the children’s weights, measured in kilograms (Boyd,
Human
Biology , 1933). The data is analyzed below.
Bivariate Fit of Volume By Weight
20
18
16
14
12
10
10 12 14 16 18 20
Weight
Linear Fit
Volume = -0.104046 + 0.9880519 Weight
Summary of Fit
RSquare
RSquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
Analysis of Variance
Source DF Sum of Squares Mean Square F Ratio
Model 1 94.099871 94.0999 2311.895
Error 16 0.651240 0.0407 Prob > F
<.0001 C. Total 17 94.751111
Parameter Estimates
Term Estimate Std Error t Ratio Prob>|t|
0.993127
0.992697
0.201749
14.72222
18

Term Estimate Std Error t Ratio Prob>|t|
Intercept -0.104046 0.311998 -0.33 0.7431
Weight 0.9880519 0.020549 48.08 <.0001
0.2
0.0
-0.2
-0.4
10 12 14 16 18 20
Weight
Distribution of Residuals
Distributions
Residuals Volume
-0.4
-0.2
0 .1 .2 .3
Moments
Mean
Std Dev
Std Err Mean upper 95% Mean lower 95% Mean
N
6.908e-16
0.1957249
0.0461328
0.0973317
-0.097332
18

Distributions
Weight
10 11 12 13 14 15 16 17 18 19
Moments
Mean
Std Dev
Std Err Mean upper 95% Mean lower 95% Mean
N
15.005556
2.3811693
0.561247
16.189683
13.821428
18
(a) Do the regression diagnostics indicate any problems with the ideal simple linear regression model holding for this data? Comment on both the residual plot and the histogram of the residuals.

For the remaining parts of the question, ignore any problems (if any) with the regression assumptions as indicated by the regression diagnostics. That is, go ahead and assume the ideal simple linear regression model holds (for the purposes of this question) even if you think the regression model should be improved.
(b) Is there strong evidence that the slope on Weight is less than 1? State hypotheses and carry out a test at the 0.05 level.
(c) Find a 95% confidence interval for the amount by which the mean Volume increases for a one kilogram increase in Weight.

(d) Predict the volume of a child who weights 25 kilograms using the simple linear regression model. Why should you not trust this prediction?
(e) A sick child weighs 14.0 kilograms (30.86 pounds). The doctor needs to decide between a treatment I, which is best for children having a volume of 13.0 cubic decimeters or less, or treatment II, which is best for children having a volume of more than 13.0 decimeters. The doctor has three options: (i) give treatment I to the child; (ii) give treatment II to the child; or (iii) measure the child’s volume. Option (iii) takes time and the child may become sicker in this time; if the doctor is confident whether treatment
I or treatment II is best, she would like to apply the appropriate treatment immediately.
What would you advise the doctor to do based on the regression analysis? Explain your answer.

2. A researcher runs a simple linear regression to estimate E(number of divorces in a city
| number of pro sports teams in city)=

0
 number of pro sports teams for a sample of
U.S. cities. The researcher finds that the least squares estimate of a test of H
0
:

1

0 vs. H
1
:

1
is positive and that

1

0 rejects the null hypothesis with a p-value of 0.005.
What is the most plausible interpretation of the results?
(a) the presence of pro sports teams causes the number of divorces to rise (perhaps husbands are spending too much time at the sports games);
(b) the high number of divorces is responsible for the presence of pro sports teams (more single men means potentially more fans at the games, making it attractive for an owner to relocate to such cities) ;
(c) the association between pro sports teams and number of divorces is due to a lurking variable (pro sports teams tend to be in larger cities hence a greater number of divorces)
(d) the observed association between pro sports teams and number of divorces is purely coincidental; it is implausible to believe the observed association could be anything other than accidental.
Answer: ______
3. Black wheateaters are small birds of Spain and Morocco. Males of the species demonstrate an exaggerated sexual display by carrying many heavy stones to nesting cavities. This 35-gram bird transports, on average, 3.1 kg of stones per nesting season!
Different males carry somewhat different sized stones, prompting a study of whether stones may be a signal of higher health status. A study was done in which the average stone mass carried by each of 21 male black wheateaters, along with the T-cell response measurements of the wheateaters was calculated. The T-cell response measurement is an indication of the strength of the bird’s immune system, with a higher T-cell response measurement indicating a stronger immune system. A simple regression analysis is shown below.

Bivariate Fit of mass By tcell
10
9
8
7
6
5
4
3
0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55
tcell
Linear Fit
Linear Fit mass = 3.9112702 + 10.165122*tcell
Summary of Fit
RSquare
RSquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
0.333634
0.298563
1.425826
7.204286
21
Analysis of Variance
Source
Model
Error
C. Total
DF Sum of Squares
1 19.339494
19
20
38.626621
57.966114
Parameter Estimates
Term
Intercept tcell
Estimate
3.9112702
10.165122
Std Error
1.112084
3.295768
Mean Square
19.3395
2.0330 t Ratio
3.52
3.08
F Ratio
9.5129
Prob > F
0.0061
Prob>|t|
0.0023
0.0061
(a) Is there strong evidence that strength of immune system is associated with how much stone mass a wheateater carries? Report an appropriate p-value and state your conclusion.

(b) Is there strong evidence that wheateaters with stronger immune systems tend to carry more stone mass? Report an appropriate p-value and state your conclusion.
(c) Is there strong evidence that wheateaters with weaker immune systems tend to carry more stone mass? Report an appropriate p-value and state your conclusion.