Statistics II-HW III
(Due 2023.08.09 23:00)
1.
(20%) Consider the following frequency table of observation on the random variable X.
Values
0
1
2
3
4
Observed Frequency
8
16
14
9
3
(a) Based on these 50 observations, is the observed distribution of X fitted a Discrete Uniform distribution?
Perform a goodness-of-fit test with   0.05 . (10%)
(b) Based on these 50 observations, is Bin(4, 0.5) an appropriate probabilistic model? Perform a goodness-offit test with   0.05 . (10%)
2. (10%) Three different design configurations are being considered for a particular component. There are
four possible failure modes for the component. The following data on number of failures in each mode for
each of the three configurations are obtained. Does the configuration appear to have an effect on type of failure?
Failure Mode
Configuration
1
2
3
4
1
2
3
20
4
10
44
17
31
17
7
14
9
12
5
State your null and alternative hypotheses. Perform a hypothesis and clearly explain the decision rule including
the rejection region. What is the conclusion? (   0.05 )
3. (20%) Given the following two independent samples
X
5.5
7.9
6.8
9.0
5.6
6.3
8.5
4.6
7.1
Y
4.3
5.9
4.9
3.1
5.3
6.4
6.2
3.8
7.5
5.8
(a) Using the large-sample Wilcoxon rank sum statistic to test whether the population distributions of these
two samples are significantly different? (10%)
(b) Using the normal approximate to calculate the p-value for test in (a). (10%)
4. (30%) To compare the property evaluations for two tax assessors, A and B. Their assessments for eight
properties are shown in the table:
Assessor
1
2
3
4
5
6
7
8
A
76.3 88.4 80.2 94.7 68.7 82.8 76.1 79.0
B
75.1 86.8 77.3 90.6 69.1 81.0 75.3 79.1
(a) Given   0.05 , use the sign test to determine whether the data present sufficient evidence to indicate that
one of the assessors tends to be consistently more conservative than the other; That is,
p  P  xA  xB   1/ 2 . (10%)
(b) Use normal approximation formula to calculate the p-value for the test in (a) and interpret it. (5%)
(c) Given   0.05 , use the Wilcoxon sign-rank test for a paired experiment to test the null hypothesis that
there is no difference in the distributions of property assessments between assessors A and B. (10%)
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(d) Compare the conclusion of the test in (a) and (c). Explain why these two conclusions are (or are not)
consistent. (5%)
5. (10%) Decide whether a difference exists in the mean number of miles driven among cars, buses, and
trucks by Kruskal-Wallis test with   0.05 .
Cars
19.9 15.3
2.2
6.8
34.2
Buses
1.8
7.2
6.5
13.3 25.4
7.2
8.3
12.0
7.0
9.5
1.1
Trucks 24.6 37.0 21.2 23.6 23.0 15.3 57.1 14.5 26.0
6.
(10%) Is the number of years of competitive running experience related to a runner’s distance running
performance? The data on nine runners, obtained from the study by Scott Powers and colleagues, are
shown in the table.
1
2
3
4
5
6
7
8
9
9
13
5
7
12
6
4
5
3
33.15 33.33 33.50 33.55 33.73 33.86 33.90 34.15 34.90
(a) Calculate the rank correlation coefficient between years of competitive running x and a runner’s finish
time y in the 10-kilometer race. (5%)
(b) Do the data provide sufficient evidence to indicate a rank correlation between y and x? Test using   0.05 .
Years of Competitive Running
10-Kilometer Finish Time (min)
(5%)
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