ST5201X

Sem 1 2023 – 2024

Due on 13 Nov 2023

Exercise 1.

STATISTICS NUS 

Tutorial 5

 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 χ2 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 χ2 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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