STAT 410/511
Exam 1
October 19, 2011
100 points
Name:
1. Researchers in Finland selected 1409 people at random from the survivors of a previous
“FINMONICA” study. They interviewed them about diet and coffee drinking, and followed them over an average of 21 years to see if they developed symptoms of dementia
and Alzheimer’s disease. The scientists report a 65% decrease in the risk of dementia for
those who drank 3 to 5 cups of coffee per day (relative to those who drank 0 to 2 cups per
day). We’ll assume that they are reporting this decrease based on “convincing evidence”
with a small p-value. What is the scope of their inference?
(12 pts)
2. Weights ( in grams) of rainbow trout captured by electrofishing on the Ruby river were analyzed based on length classes (length cut into 25mm intervals) and the residual diagnostic
plots are shown.
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−100
●
● 11145
100
150
200
Fitted values
250
300
350
●
●
●
●
● ●
●
●
●
● ●
●
●●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−2
−1
0
1
Theoretical Quantiles
2
3
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● 11145
7214
● 6853
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−3
●
●
●
●
●
●
● 11145
50
2.0
4
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
1.5
●
●
●
●
●
1.0
●
●
●
●
Standardized residuals
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0.5
●
●
●
0.0
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●●
2
●
●
●
−4
0
●
●
●●
●●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
−50
Residuals
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
Scale−Location
6853
7214
● ●
●
0
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
7214
● 6853
●
Standardized residuals
50
●
Normal Q−Q
−2
100
Residuals vs Fitted
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
50
100
150
200
250
300
350
Fitted values
Discuss any violations of the assumptions of ANOVA visible in the plots.
(12 pts)
Stat 410/511 Midterm Page 2
3. Barrick and Showers collected data on oxygen isotopic composition for 12 bones (each
measured three or more times) from a single Tyrannosaurus rex specimen. They wanted
to see if the means are equal for the 12 bones because that helps answer the question of
dinosaurs being warm or cold blooded.
(a) Using the model yij = µi + ij for i = 1, . . . , 12 and j = 1, . . . , ni , express the null
and alternative hypotheses in terms of model parameters. (The usual hypotheses for
an ANOVA setting).
(8 pts)
12.0
(b) The data are plotted below and we have a partial anova table.
11.0
11.5
●
1
2
3
4
5
6
Df
7
8
9
Sum Sq Mean Sq F value
6.07
40
2.97
bone
Residuals
Total
Fill in one blank at a time below:
10
11
12
Pr(>F)
0.0001
i. Df for bone group in line 1.
(2 pts)
ii. Total Df
(2 pts)
iii. Total Sum Sq
(2 pts)
iv. Mean Sq for line 1
(2 pts)
v. Mean Sq for line 2
(2 pts)
vi. F value for line 1
(2 pts)
vii. Under one hypothesis, we know the distribution of F. Which hypothesis, and
what is that distribution?
(6 pts)
Stat 410/511 Midterm Page 3
(c) State your conclusions based on the above F test.
(8 pts)
(d) The bones can be subdivided into four groups according to proximity to the body
core. The warm/cold blooded question involves differences between these four groups.
i. We want to use an extra sum of squares F test to compare the four groups model
to a model with one mean. The SSE for a four means model is 7.16. Find the
Extra Sum of Squares and the top of the fraction.
(8 pts)
ii. Compute the bottom of the fraction (show work).
(5 pts)
iii. Compute the F statistic and give its degrees of freedom (show work):
(5 pts)
iv. The p-value is 0.010. State your conclusion.
(5 pts)
(e) Is there a problem with measuring each bone multiple times? Discuss in terms of the
assumptions for ANOVA.
(5 pts)
Stat 410/511 Midterm Page 4
4. Consider two-sample t-procedures applied to log transformed data.
(a) Draw a side-by-side boxplot of data which need log transformation. Describe two
characteristics we observe in such a plot which tell us logs are needed.
(6 pts)
(b) In the cloud seeding example of Sleuth §3.5, we estimated that seeding was associated
with an increase of 1.14 in the log scale (SE = 0.45) with a 95% confidence interval
for the difference in log means of (0.24, 2.05). Interpret this interval on the original
scale (in acre-feet).
(4 pts)
(c) What do we mean when we say we have 95% confidence in an interval?
(4 pts)