LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

  FIFTH SEMESTER – APRIL 2004

ST 5501/STA 506 – TESTING OF HYPOTHESIS

06.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION – A

 

Answer ALL questions                                                                                (10 ´ 2 = 20 marks)

 

1. Define a simple hypothesis and a composite hypothesis.
2. Let X₁, X₂,…, X_n be a random sample from N (m, s²).

Write the distributions of   i)            ii) .

3. Define uniformly most powerful critical region.
4. Explain Type – I error and Type – II error.
5. Explain the likelihood ratio principle.
6. What do you mean by non- parametric methods?
7. When do we need the randomised test?
8. Find the number of runs in the sequence.

x yyy   xxx y  x y xxx  yy  xxxxx

9. Explain the term confidence interval.
10. What is a p – value?

 

SECTION – B

 

Answer any FIVE questions                                                                        (5 ´ 8 = 40 marks)

 

11. Let X have pdf of the form f (x, q) = q x^q-1, 0 < x < 1, zero elsewhere, where q Î {q ½q = 1,2}.  To test H_o: q = 1 vs H₁ : q = 2, a random sample of size 2 is chosen.  The critical region is C = { (x₁, x₂) ½  < x₁ x₂}.  Find Type I error and Type II error.
12. Verify whether UMPT exists for testing

H_o: q = q^‘ Vs H₁: q ¹ q^‘

when the random sample X₁, X₂, …, X_n is from N (q , 1).

13. Explain Wilcoxon’s Test.
14. The theory predicts the proportion of beans in the 4 groups A, B, C and D should be

9 : 3 : 3 : 1.  In an experiment among 1600 beans, the number in the 4 groups were 882,

313, 287, 118.  Does the experimental results support the theory?

15. Explain how will you test for regression coefficients b and a in

y_i = a + b (c_i – ),    i = 1, 2, … n

16. Explain the t-test for equality of means of two independent Normal populations.
17. In a random sample of 500 men from a particular district of Tamil Nadu, 300 are found to be smokers. In one of 1000 men from another district, 550 are smokers.  Do the data indicate that the two districts are significantly different with respect to the prevalence of smoking among men?
18. Derive the distribution of number of runs.

SECTION – C

 

Answer any TWO questions                                                                        (2 ´ 20 = 40 marks)

 

19. a) State and prove Neyman – Pearson lemma.          (10)
20. b) Explain monotone likelihood ratio property (MLR) and its use in testing the

hypothesis.                                                                                                                  (10)

 

20. a) Derive the likelihood ratio test for testing the equality of two variances of two normal

populations N (q₁, q_3­) and N (q₂, q_4­), q₁, q₂ unspecified.                                             (12)

1. b) Two independent samples of 8 and 7 respectively had the following values of the

variables.

Sample I          9          11        13        11        15        9          12        14

Sample II        10        12        10        14        9          8          10

Do the population variances differ significantly?                                                       (8)

 

21. a) Explain Man-Whitney – Wilcoxon Test.           (10)
22. b) Explain Sign-Test (10)

 

22. a) Test the hypothesis that there is no difference in the quality of the 4 kinds of tyres A,

B, C and D based on the data given below:

Tyre Brand
	A	B	C	D
Failed below 40,000 kms	26	23	15	32
Lasted from 40,000 to 60,000 kms	118	93	116	121
Lasted more than 60,000 kms	56	84	69	47

1. b) Let X₁, X₂, …, X_n be a random sample from N (q, 100). Find n and c if

H_o­: q = 75 vs H₁ :  q = 78.  Given P [ X Î C ½H_o] = .05   and P [X Î C ½H₁] = .90.

C = { (x₁, x₂, …,x_n) ½  ≥ c}  is the Best critical region.

 

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