THE UNIVERSITY OF NEW SOUTH WALES
DEPARTMENT OF STATISTICS
Exercises for MATH3811, Statistical Inference
One Sample Problems. Two Sample Problems
1. In a study of drug abuse in a suburb A it was found that the median I.Q. of arrested abusers who
were 16 years of age or older was 107. In order to test whether the median I.Q. of arrested abusers
aged 16 or more in another suburb B is also 107, the I.Q.’s of 12 such persons from the suburb B
were determined and are given below:
108, 111, 99, 127, 109, 100, 90, 94, 135, 104, 119, 117
a) Carry out an appropriate sign test.
b) Give a non-parametric point estimate and find 95% confidence limits for the median I.Q. of drug
abusers aged 16 or more in the suburb B.
2. The following table gives the level of the chemical serotonin in the blood of a patient before and after
injecting 2.5mg of the drug reserpine.
Patient
1
2
3
4
5
6
7
8
9
10
Level before 0.61 0.59 0.68 0.67 0.70 0.56 0.64 0.59 0.64 0.64
Level after 0.56 0.46 0.59 0.53 0.79 0.50 0.54 0.52 0.72 0.60
a) Carry out a sign test of the hypothesis that there is no difference in the serotonin level of a
patient before and after injecting reserpine. What assumptions need to be made to carry out
the test?
b) Also carry out a Wilcoxon signed rank test of the same hypothesis stating the assumptions
needed to be made for the test to be valid.
3. A psychological experiment was conducted to compare the length of response time (in seconds) for
two different stimuli. In order to remove natural person-to person variability in the responses both
stimuli were applied to each of nine subjects, thus permitting an analysis of the difference between
stimuli within each person.
Subject
1
2
3
4
5
6
7
8
9
Stimulus 1
9.4 7.8 5.6 12.1 6.9 4.2
8.8 7.7 6.4
Stimulus 2 10.3 8.9 4.1 14.7 8.7 7.1 11.3 5.2 7.8
Use the sign test to determine whether sufficient evidence exists to indicate a difference in mean
response for the two stimuli. State the actual level of significance of your test.
4. In a controlled environment laboratory, ten men and ten women were tested to determine the room
temperature they found to be most comfortable. The results were as follows:
Men: 74 72 77 76 76 73 75 73 74 75
Women: 75 77 78 79 77 73 78 79 78 80
Stating the assumptions you need to make, carry out a Wilcoxon test of the hypothesis that the
average comfortable temperature is the same for men and women. State the exact level of significance
of your test procedure.
5. In an annual survey to determine whether state pay scales were commensurate with private sector
salaries, government and private workers were matched as closely as possible (with respect to the
type of job, educational background, years experience, etc.) and the salaries of the matched pairs
were obtained. The data in the table are the annual salaries for 12 such matched pairs:
1
Private:
Government:
12500
11750
22300
20900
14500
14800
32300
29900
20800
21500
19200
18400
15800
14500
17500
17900
23300
21400
42100
43200
16800
15200
14500
14200
Test the hypothesis H0 of equal pay versus the alternative that government workers are generally
paid less than their counterparts in the private sector using the signed rank test.
6. The recorded high temperature in a Florida resort town for each of 10 consecutive days during the
month of January of this year is compared with the historical average high for the same days in
previous years and noted as either above historical average (A) or below historical average (B). For
the data A A B A B B A A A B, test the null hypothesis of random direction of deviation from
average high temperature against the alternative of nonrandomness, using level 0.05.
7. A random sample of seven boys was selected from a group of boys in a senior class who lived on
farms. Another random sample of nine boys was selected from a group of boys in the same class who
lived in town. A test was devised and given to these sixteen boys to see whether farm boys in general
were more physically fit than town boys. The scores of the farm boys (Xi ) and the town boys (Yi )
are as follows:
Xi (Farm boys): 14.8 5.6
9 10.6 12.9 11.4
2.7
Yi (Town boys):
2.4 7.9 10.6
9.1 10.6
5.0 18.6 5.6 4
Carry out a Wilcoxon test of the hypothesis that there is no difference in the physical fitness of farm
boys and town boys.
8. A report on the histamine level for a sample of 9 individuals classified as allergics and another sample
of 13 individuals classified as nonallergics is given below:
Allergics 67.6 39.6 1651
100 65.9 1112
31 102.4 64.7
Nonallergics 34.3 27.3 35.4 48.1
5.2 29.1 4.7
41.7
48 6.6 18.9 32.4 45.5
Does the data indicate at level 1% that there is a difference in true average histamine level between
allergics and nonallergics? (Since both sample sizes exceed 8, use the normal approximation to the
Wilcoxon statistic.)
9. X and Y are two continuous random variables with cumulative distribution functions F (t) and
F (t − θ) respectively. Let X1 , X2 , . . . , Xm and Y1 , Y2 , . . . , Yn denote random samples from the two
populations and suppose P (Y < X) = ω.
R∞
a) Prove that ω = −∞ F (x − θ)f (x)dx. Hence verify that ω = 0.5 when θ = 0.
b) Consider the Mann-Whitney statistic
U=
m X
n
X
h(Xi − Yj ), h(u) = I(0,∞) (u).
i=1 j=1
Find E(U ) and an unbiased estimator of ω.
10. Twenty four “alcohol-dependent” male patients at an alcohol treatment center were randomly assigned to two groups. The control group patients were given a traditional treatment program. The
treatment group patients were given the traditional treatment program plus a class in social skills
training (SST). After being discharged from the program, each patient reported- in 2-week intervalsthe quantity of alcohol consumed, the number of days prior to his first drink, the number of sober
days, the days worked, the times admitted to an institution, and the nights slept at home. Reports
were verified by other sources (wifes or family members). (Such data can be unreliable !) One patient
in the SST group, discovered to be an opiate addict, disappeared after discharge and submitted no
reports. The remaining 23 patients reported faithfully for a year. The results for alcohol intake (cl
of pure alcohol for 1 year) are given below:
Control: 1042, 1617, 1180, 973, 1552, 1251, 1151, 1511, 728, 1079, 951, 1319.
SST: 874, 389, 612, 798, 1152, 893, 541, 741, 1064, 862, 213.
Test H0 of equal alcohol intakes versus the alternative that the SST group tends to have lower alcohol
intakes at α = 0.05. Use the exact Wilcoxon rank sum test and the large-sample approximation to
it. Report and compare the results.
2
Answers
1) a) Accept H0 ; b) point estimate: 108.5; (99,119) (96.2%) confidence limits )
2) Accept H0 (ie. decide there is no difference)
3) Accept H0 at 5% level. 4) Reject H0
5) Reject H0 of equal pay 6) Do not reject the hypothesis of randomness
7) Accept H0 8) Z statistic= 3.17 Reject H0 at 1% level.
9) b) mnω, U/(mn) 10) n = 12, m = 11, exact: W = 81, normal approximation= −3.14, reject H0 .
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