Sample Final Exam from Spring 2011
Solution to Problems 2, 3, and 4
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.
Response:
percentage of unburned popped kernels
Conditions: Wattage (2 levels)
Time (3 levels)
Material:
packages of microwave popcorn
Because different brands and styles vary in terms of size and contents if
different brands and styles are included in the experiment, the material will
not be uniform. If only one brand and style is uses (controlling for brand
and style) the conclusions will not cover several different brands and styles of
microwave popcorn. To accommodate this, form blocks by sorting packages
by brand and style. With 6 treatment combinations, there can be 10 blocks.
Source
Wattage
Time
Watt*Time
Block
Error
C. Total
df
1
2
2
9
45
59
1
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.
Response:
eye focus time
Conditions: Gender (2 levels) – observational factor
Alcohol (4 levels) – experimental factor
Material:
young adults
Young adults will vary a great deal in terms of their height, weight and
tolerance to alcohol so a completely randomized design is not the best design.
Gender cannot be assigned and it may be hard to make homogeneous blocks
containing both men and women (hard to get men and women the same
height, weight and tolerance). This experiment should be done as a split plot
design. Whole plot factor is gender, with 20 males and 20 females. For the
subplot we could:


Form blocks by sorting on height/weight/tolerance – 5 groups of 4
males and 5 groups of 4 females.
Form blocks by reusing each adult
The subplot factor is amount of alcohol.
Block by sorting
Source
Gender
Groups[Gender]
Alcohol
Gender*Alcohol
Error
C. Total
df
1
8
3
3
24
39
Block by reusing
Source
df
Gender
1
Groups[Gender]
38
Alcohol
3
Gender*Alcohol
3
Error
114
C. Total
159
2
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.
Response:
corn yield (bushels per acre)
Conditions: Depth (2 levels)
Date (3 levels)
Material:
Fields
This is a split plot design. The whole plot is a field and the whole plot factor
is the planting depth. The subplot is a third of a field and the subplot factor
is the date of planting. Each depth is assigned at random to 12 fields,
completely randomized design. The three dates are assigned at random to
each field, making each field a block which is subdivided to accommodate the
three dates, in a randomized complete block design.
Source
Depth
Fields[Depth]
Date
Depth*Date
Error
C. Total
df
1
22
2
2
44
71
3
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.
Response:
plasma protein level
Conditions: Gender (2 levels) – observational factor
Fasting (3 levels) – experimental factor
Material:
turtles
This looks like it should be a split plot design because gender is an
observational factor that cannot be assigned at random. However, there is
no mention of blocking. Turtles are not sorted, within males and females, to
make them more uniform nor are they reused. What we end up with is 6
groups of 10 turtles. The groups correspond to the combination of gender
and fasting:
Male/None, Male/10 days, Male/20 days
Female/None, Female/10 days, Female/30 days
Source
Gender
Fasting
Gender*Fasting
Turtles[Gender,Fasting]
C. Total
df
1
2
2
54
59
The only source of variation is differences among turtles treated the same
(the same gender assigned the same fasting level). This is the error term for
a completely randomized design.
4
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?
A whole plot is a rectangular plate.
b) [2] What is the whole plot factor?
The whole plot factor is the type of acid, Rosmarinic acid or Ursolic acid.
c) [2] What is the sub plot?
A sub plot is a well in the rectangular plate.
d) [2] What is the sub plot factor?
The sub plot factor is the concentration of acid, 1, 2, 5, 10, 20 and 50 M .
e) [4] The two compounds are statistically different. Give the appropriate F statistic
that supports this result.
𝑭=
𝑴𝑺𝑪𝒐𝒎𝒑𝒐𝒖𝒏𝒅
𝟒𝟓. 𝟖𝟑𝟓
=
= 𝟗𝟔. 𝟗𝟐𝟑
𝑴𝑺𝑷𝒍𝒂𝒕𝒆[𝑪𝒐𝒎𝒑𝒐𝒖𝒏𝒅] 𝟎. 𝟒𝟕𝟐𝟗
Note: Although not necessary because there are only two compounds, if we
were to compute a value for the LSD it would use MSPlate[Compound] as the
MSError.
𝟐
𝑳𝑺𝑫 = 𝟐. 𝟏𝟒𝟒𝟕𝟗√𝟎. 𝟒𝟕𝟐𝟗√ = 𝟎. 𝟑𝟎
𝟒𝟖
5
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
Estimated Effect
1.59
2
12.00
1.27
5
11.56
0.83
10
10.58
–0.15
20
9.26
–1.47
50
8.66
–2.07
Overall
10.73
g) [5] Compute the value of the HSD (use q* = 2.93014) and indicate which
concentration means have statistically significant differences.
𝟐
𝑯𝑺𝑫 = 𝟐. 𝟗𝟑𝟎𝟏𝟒√𝟎. 𝟐𝟓𝟒𝟐√ = 𝟐. 𝟗𝟑𝟎𝟏𝟒(𝟎. 𝟏𝟕𝟖𝟐𝟓𝟓) = 𝟎. 𝟓𝟐
𝟏𝟔
Concentrations 1 and 2 are not statistically different, difference in means =
0.32 < 0.52. Concentrations 2 and 5 are not statistically different, difference
in means = 0.44 < 0.52.
All other comparisons between pairs of
concentrations have statistically significant differences, difference in means >
0.52.
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.
6
Both acids have a decreasing trend between the mean Ln(Cell) and
concentration. As concentration increases the log of the number of cells
tends to decrease. Rosmarinac acid starts off lower and decreases more
slowly. Ursolic acid starts off higher and decreases more rapidly. At lower
concentrations there is a big difference between the means for the two acids.
For higher concentrations there is very little difference between the means
for the two acids.
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
7