ST5201X
STATISTICS NUS
Sem 1 2024 – 2025
Due on 10 Nov 2024 Tutorial 5
Exercise 1.
Find the exact distribution for the Wilcoxon rank sum statistics W2 and U2 when n1 = 4, n2 = 2. Use
this to find P(U2 ≤ 3) and P(U2 ≥ 8).
Exercise 2.
Tests are conducted on two brands of batteries A and B to see which one lasts longer. The following
tables show their lifetimes (in hours) of continuous use.
Brand A : 11.7, 10.0, 10.8, 11.1, 12.9.
Brand B : 11.5, 12.8, 13.8, 13.6, 15.5, 12.4.
Test whether Brand B battery lasts longer or not at significance level 0.05 using
(a) The Wilcoxon rank sum test,
(b) Normal approximation to the Wilcoxon rank sum test
(c) 2-sample t-test. (Assuming the data is from normal distribution.)
(d) Did you reach the same conclusions with the above three approaches ? If not, explain.
Exercise 3.
The following data gives miles per gallon of two different models of cars made by two different manufac
turers.
Before, x 17.2 21.6 19.5 19.0 22.0
After, y 18.3 20.8 20.9 21.2 22.7
We assume that the treatment effect (i.e. difference) follows a shift model G(t) = F(t + ∆)
(a) Find an estimate of ∆.
(b) Find a two-sided confidence interval for ∆ which is closest to 90% level (hint : use table
A4 for hand calculation).
Exercise 4.
We draw 5, 5, 7 samples from 3 populations respectively. The results are
5, 4, 6, 4, 6
X1j ∼ F1.
7, 3, 5, 6, 5
X2j ∼ F2.
2, 3, 3, 1, 2, 1, 1
X3j ∼ F3.
Check the claim that there is no difference in the three populations at significance level 0.01 using
the Kruskal-Wallis test (using χ
2 approximation).
Exercise 5.
The following data gives blood cholesterol levels of men in three different socioeconomic groups labeled
I, II, III with I being the high end.I : 403, 311, 269, 336, 259
II : 312, 222, 302, 420, 420, 386, 353, 210, 286, 290
III : 403 244 353 235 319 260
We want to test whether there is a significant difference at level 0.1 between the mean blood cholesterol
of three groups. using
(a) The Kruskal-Wallis test (using χ
2 approximation).
(b) ANOVA method (i.e. F-ratio test). (Assuming the data are all from normal distributions.)
(c) Did you reach the same conclusions with the above two approaches ? If not, explain.
Exercise 6.
If p = 2 and under H0 : F1 = F2, prove that the KW statistic T = (W2 − EW2)
2/varW2. (Recall : W2 is
the sum of ranks in sample 2 defined in Lecture 10-3).
Your submission should be sent in PDF format to Canvas before the deadline.
The name of your submission is restricted to the format as ID.pdf, in which ID is
your Matric No.
You are only allowed to submit one PDF for each tutorial
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