One Way ANOVA
NAME: _______________________________________
Chapter 12
Comparing Many Treatments
1.
Analysis of Variance gets its name from the fact that it is a statistical technique for
assessing whether the variances of several groups or populations are equal.
(a)
True
(b)
False
2.
One assumption for ANOVA is that the sample variances for each group must be the
same.
(a)
True
(b)
False
3.
Fill in the blank. One assumption for ANOVA is that the distribution of the populations
from which the samples come is the
distribution.
(a)
t
(b)
F
(c)
normal
(d)
uniform
4.
Fill in the blank. If all the assumptions for ANOVA hold, the test-statistic calculated in
ANOVA will follow the
distribution.
(a)
t
(b)
F
(c)
normal
(d)
uniform
5.
Fill in the blank. The null hypothesis for ANOVA states that all I means are equal to each
other. The alternative hypothesis states that
.
(a)
all I means are different from each other
(b)
none of the I means are equal to each other
(c)
not all I means are equal to each other
(d)
6.
Fill in the blank. The basic concept underlying ANOVA is comparing the variation
between the sample means (MSB) with the random variation of the observations within
the samples (MSW). To have more support for the conclusion that the population means
are different, the MSB should be
as compared to the MSW.
(a)
large
(b)
small
7.
Fill in the blank. An ANOVA will be used to test if the average rental costs for 1bedroom apartments for 4 cities are equal. A random sample of 16 1-bedroom apartments
from each city will be selected. Respectively, the degrees of freedom for the numerator
and denominator of the F-statistic are _____________.
(a)
4 and 16
(b)
4 and 64
(c)
3 and 16
(d)
3 and 60
(e)
none of the above answers is correct
Chapter 12
Questions 8 through 12 are based on the following ANOVA table.
SOURCE
SS
BETWEEN
(Error)
TOTAL
MS
F
2
(Groups)
WITHIN
DF
480
30
660
8.
Fill in the blank. The number of populations being compared is _______ .
(a)
1
(b)
2
(c)
3
(d)
4
(e)
none of the above answers is correct
9.
Fill in the blank. The total number of observations, n, is _______ .
(a)
16
(b)
18
(c)
19
(d)
30
(e)
none of the above answers is correct
10.
Fill in the blank. The mean square between groups, MSB, is _______ .
(a)
90
(b)
30
(c)
180
(d)
480
(e)
none of the above answers is correct
11.
Fill in the blank. The observed F-test statistic value is _______ .
(a)
2
(b)
3
(c)
30
(d)
16
(e)
none of the above answers is correct
12.
Fill in the blank. The p-value for the test of equality of the means is _______ .
(a)
0.0783
(b)
0.1565
(c)
0.10
(d)
less than 0.05
Questions 13 through 15 are based on the following scenario.
A clinical psychologist wished to compare three methods for reducing hostility levels in
university students. A certain test (HLT) was used to measure the degree of hostility. A
high score on the test indicated great hostility. Eleven students obtaining high and nearly
equal scores were used in the experiment. Four were selected at random from among the
11 problem cases and treated with method 1. Four of the remaining 7 were selected at
random and treated with method 2, and the remaining were treated with method 3. All
treatments were continued for a one-semester period. Each student was given the HLT
test at the end of the semester, with results and output shown below. Assume that the
distribution for HLT scores for the three methods are normally distributed with equal
population variance.
Test Scores for Method 1:
Test Scores for Method 2:
Test Scores for Method 3:
80
70
63
92
81
76
87
78
70
83
74
13. Complete the ANOVA output.
14.
Give the appropriate hypotheses for assessing if there is a difference among the mean
scores for the three methods.
15.
State your conclusions regarding the differences between methods based on the using an
overall significance level of 5%.
Chapter 12
Questions 16 through 18 are based on the following scenario.
In July, 1995, the magazine "Car and Driver" conducted speed tests of five supercars
from five different countries: 1 = Japan (Acura NSX-T), 2= Italy (Ferrari F355), 3 = UK
(Lotus Esprit S4S), 4 = Germany (Porsche 911 Turbo), 5 = USA (Dodge Viper RT/10).
Car and Driver recorded the top speeds (mph) achieved by these cars using as much
distance as necessary. Six measurements were made on each car, three in each direction
to cancel any grade or wind factors at the test facility. The analysis of variance results for
comparing these speed measurements is provided below.
- - - - Variable
By Variable
Source
Between Groups
Within Groups
Total
MPH
CAR
D.F.
4
25
29
O N E W A Y
- - - - -
car type
Analysis of Variance
Sum of
Mean
Squares
Squares
1455.1847
363.7962
362.5650
14.5026
1817.7497
F
Ratio
25.0849
F
Prob.
.0000
16.
What null and alternative hypotheses are being tested by the F-Ratio in the Analysis of
Variance table shown above?
17.
What assumptions are required for the F-test in the Analysis of Variance table to be
valid?
18.
Find The F-ratio and p-value. At a 5% significance level, what is your decision?
(a)
Fail to Reject H0
(b)
Reject H0
Questions 48 through 53 are based on the following scenario.
The Silver Family is going to have an addition built to their house. They need to decide
which contractor is going to oversee the project. They searched for somebody in the
following 3 categories: general contractors, carpenters, and handymen. They visited some
people from each category and obtained bids from each one of them. The bids (in
thousand dollars) is shown below.
General contractors: 51
Carpenters:
43
Handymen:
34
47
36
27
48
38
39
39
42
33
Complete the table .
19.
Give the appropriate hypotheses for assessing if there is a difference among the mean bids
for the three categories.
20.
Test the hypotheses of question 19. Report the value of the observed test statistic, the pvalue, and your decision using a 1% significance level.
21.
State your conclusions regarding the differences between categories.
Questions 22 through 26 are based on the following scenario.
Chapter 12
Three neighbors are having a dispute over whose dog runs the fastest. They decide to do a
little experiment. Each dog will run a short distance three times and their times will be
recorded. Neighbor Warner knows a little about statistics and decides to use an ANOVA
test. The times (in seconds) for the three dogs are listed below.
Jake = 1
11.0
12.9
12.1
Rover=2
13.5
10.8
14.1
Molly=3
9.9
9.8
9.5
22.
Besides assuming that the running times of the dogs are normally distributed and that the
runs are independent, what else should Neighbor Warner assume?
23.
Calculate the MSB. What does this represent?
24.
The MSW turns out to be 2.60. What is the value of the test statistic?
25.
Based on the test statistic from question 21, what would you tell the 3 neighbors?
26. Determine which Runner(s) differ?
Jake & Rover
Jake & Molly
Rover &Molly
They are all differernt from each others
Multiple Comparisons
Tukey
(I)
(J)
95% Confidence Interval
Runner Runner Mean Difference
_label
_label
1
2
-.80000
.94790
.692
-3.7084
2.1084
3
2.26667
.94790
.117
-.6418
5.1751
1
.80000
.94790
.692
-2.1084
3.7084
3
3.06667*
.94790
.041
.1582
5.9751
1
-2.26667
.94790
.117
-5.1751
.6418
2
-3.06667*
.94790
.041
-5.9751
-.1582
2
3
(I-J)
Std. Error
Sig.
*. The mean difference is significant at the 0.05 level.
Lower Bound
Upper Bound