Download Statistics 402B Final Exam Name: __________________________

Statistics 402B
May 5, 2011
Final Exam
Name: __________________________
INSTRUCTIONS: Read the questions carefully and completely. Answer the questions
and show work in the space provided. This is the only work that I will look at. Partial
credit will not be given if work is not shown. Be sure to answer all questions within the
context of the problem. Refer to the computer printout and graphs provided when
appropriate. If the computer printout has the answer use it, you do not have to calculate
by hand. Pace yourself. Do not spend too much time on any one problem. Point values
for each problem are given. There are a total of 85 points on the exam.
1. [40 pts] In the spring a young man’s (or woman’s) fancy may turn to thoughts of ...
golf. Two students at Napier University in Scotland (where else) performed an
experiment to investigate the effects of several factors on the distance (in yards) a
golf ball travels. The factors and data are given below. The experiment was run as a
completely randomized design with one observation at each treatment combination.
Factor
A: Ability (handicap)
B: Teeing
C: Club
D: Ground
A
Ability
–1
+1
–1
+1
–1
+1
–1
+1
–1
+1
–1
+1
–1
+1
–1
+1
B
Teeing
–1
–1
+1
+1
–1
–1
+1
+1
–1
–1
+1
+1
–1
–1
+1
+1
–1
lower (8)
no tee
wood
soft
C
Club
–1
–1
–1
–1
+1
+1
+1
+1
–1
–1
–1
–1
+1
+1
+1
+1
D
Ground
–1
–1
–1
–1
–1
–1
–1
–1
+1
+1
+1
+1
+1
+1
+1
+1
+1
higher (4)
tee
metal
hard
Y
Distance
153
195
211
232
160
204
222
236
183
260
242
285
200
264
276
301
a) [1] What is the response?
1
b) [2] What are the conditions?
c) [3] This experiment was run as a completely randomized design. Explain how
you would have done the randomization. Be specific.
d) [8] Plots of the main effects are given below. Comment on the apparent effects of
each of the factors.
2
e) [6] Plots of some of the interactions are given below and on the next page.
Comment on the apparent interaction between factors Ability and Tee and Ground
and Tee. In your comments you cannot use the word “parallel.”
3
Below is a table of estimated full effects, labeled by name.
Name
Mean
A
B
AB
Full Effect
226.50
41.25
48.25
–15.50
Name
C
AC
BC
ABC
Full Effect
12.75
–4.50
3.50
–1.75
Name
D
AD
BD
ABD
Full Effect
49.75
11.00
1.00
–2.75
Name
CD
ACD
BCD
ABCD
Full Effect
5.00
–3.25
3.75
0.50
f) [4] Below is a normal probability plot of full effects. Label with the name those
effects that appear to be important.
4
g) [3] Are there factors that can be dropped? If so, which ones? Explain briefly.
h) [4] Below are effect tests for the main effects and 2-way interactions. According
to this output, what effects and interactions are statistically significant?
Effect Tests
Source
A
B
A*B
C
A*C
B*C
D
A*D
B*D
C*D
Nparm
1
1
1
1
1
1
1
1
1
1
DF
1
1
1
1
1
1
1
1
1
1
Sum of Squares
6806.2500
9312.2500
961.0000
650.2500
81.0000
49.0000
9900.2500
484.0000
4.0000
100.0000
F Ratio
239.6567
327.8961
33.8380
22.8961
2.8521
1.7254
348.6004
17.0423
0.1408
3.5211
Prob > F
<.0001*
<.0001*
0.0021*
0.0049*
0.1521
0.2460
<.0001*
0.0091*
0.7228
0.1194
i) [4] Using only those factors and interactions that are statistically significant give
the prediction equation for distance and use this equation to predict the distance
for a higher ability golfer using a metal club and a tee on hard ground?
5
j) [5] You are doing multiple tests in h). If each test uses a significance level of
0.05 then the chance of making at least one Type I error in the set of 10 tests is
much higher than 0.05. What could you do to avoid this problem but still do all
10 tests? Explain briefly. How will this change your results in h)?
2. [10 pts] For each of the following situations explain what design you would use to
accomplish the stated purpose. Be sure to support your choice of design. If you
chose a design that involves blocking be sure to indicate how you will make your
blocks.
a) [5] We wish to investigate how microwave wattage (1000 watt microwave or
1500 watt microwave) and time (3 minutes, 3.5 minutes or 4 minutes) affect the
percentage of unburned popped kernels in microwave popcorn. For a given brand
and style of popcorn, microwave popcorn comes in sealed packages with the same
weight of popcorn in each. The size and contents of different brands and styles
will vary. You have enough resources to purchase 60 packages and you want
your conclusions to cover several different brands and styles of microwave
popcorn.
6
b) [5] An ophthalmologist wishes to see how eye focus time in young adults is
affected by the amount of alcohol consumed and whether the gender of the
individual has an influence. The amounts of alcohol are none, 2 ounces, 4 ounces
and 6 ounces consumed in a 20-minute period. The investigator is able to
randomly select 20 young adult females and 20 young adult males from all those
young adults willing to participate in the experiment. It is known that body
composition (height and weight) and tolerance to alcohol varies a lot among
young adults.
3. [10 pts] For each of the following situations,
 Give the design that was used (completely randomized, randomized complete
block, Latin square, split plot/repeated measures).
 Give a partial ANOVA table listing all sources of variation and degrees of
freedom.
a) [5] An experiment is performed to study the effect of depth of planting (½ inch, 1
inch) and date of planting (April 30, May 7, May 14) on corn yields. There are 24
fields available for the experiment. Twelve of the fields are assigned at random
to each of the planting depths. On one third of each field corn seed is planted
April 30, on another third of each field corn seed is planted on May 7 and on the
final third of each field corn seed is planted on May 14. Which third is planted on
each day is determined at random for each field. All the fields are harvested on
the same day in late October and the yield of corn (bushels per acre) is recorded.
7
b) [5] An experiment is done using 60 turtles. Thirty of the turtles are male and
thirty of the turtles are female. The male turtles are randomly divided into 3
groups of 10 and the female turtles are randomly divided into three groups of 10.
One group of males and one group of females has their plasma protein measured
while they are well fed (no fasting). One group of males and one group of females
has their plasma protein measured after ten days of fasting. One group of males
and one group of females has their plasma protein measured after twenty days of
fasting. The researcher wishes to know if there are differences between the
genders. The researcher also wants to know if there are differences among the
fasting times and if there is any interaction between gender and fasting time.
4. [25 pts] Rosmarinic acid is an antioxidant found in herbs of the mint family. Ursolic
acid, found in apples and other fruits and herbs, is capable of inhibiting growth of
some cancer cells. An experiment is performed to see the effects of concentration
levels of Rosmarinic acid, RA and Ursolic acid, UA on cells. Sixteen rectangular
plates, each with six wells, will be used. The experimenter wants to avoid crosscontamination of compounds so only one compound will be used on a plate. There
are six concentration levels: 1, 2, 5, 10, 20 and 50 M . The experiment is done as a
split plot design. Each acid is randomly assigned to eight of the rectangular plates.
The six concentration levels are randomly assigned to the six wells within each plate
for each compound. The response is the logarithm of the number of viable cells at the
end of the incubation time. Refer to the JMP output “Effects of acids on cell
numbers.”
a) [2] What is the whole plot?
b) [2] What is the whole plot factor?
c) [2] What is the sub plot?
d) [2] What is the sub plot factor?
8
e) [4] The two compounds are statistically different. Give the appropriate F statistic
that supports this result.
f) [3] There are some statistically significant concentration effects. Below are the
means for the six concentrations. Calculate the estimated effects of each
concentration level.
Concentration
1
Mean
12.32
2
12.00
5
11.56
10
10.58
20
9.26
50
8.66
Overall
10.73
Estimated Effect
g) [5] Compute the value of the HSD (use q* = 2.93014) and indicate which
concentration means have statistically significant differences.
9
h) [5] There is a statistically significant interaction between Compound and
Concentration. Below is the interaction plot. Describe the plot and comment on
the nature of the interaction.
15
RA
UA
Ln(Cell) LS Means
10
5
1
2
5
10
20
50
Concentration
Effects of Acids on Cell Numbers
Response Ln(Cell)
Summary of Fit
RSquare
RSquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
0.935289
0.912178
0.504229
10.73068
96
Analysis of Variance
Source
DF Sum of Squares
Model
25
257.23058
Error
70
17.79726
C. Total
95
275.02784
Effect Tests
Source
Compound
Concentration
Compound*Concentration
Plate[Compound]
Mean Square
10.2892
0.2542
DF Sum of Squares
1
45.83500
5
181.18668
5
23.58829
14
6.62061
F Ratio
40.4695
Mean Square
45.83500
36.23734
4.71766
0.47290
Prob > F
<.0001*
F Ratio
10