LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AB 14

FIFTH SEMESTER – NOV 2006

ST 5501 – TESTING OF HYPOTHESIS

(Also equivalent to STA 506)

 

 

Date & Time : 27-10-2006/9.00-12.00   Dept. No.                                                       Max. : 100 Marks

 

 

PART – A

Answer ALL the questions.                                                        10 X 2=20 marks

 

1. Define Type I and Type II errors.
2. Give an example each for a simple and composite hypothesis.
3. When do you say that a critical region is uniformly best?
4. Define level of significance and power function of a test.
5. Provide a testing problem for which no uniformly best critical region exists.
6. When a distribution is said to belong to an exponential family?
7. When a” Likelihood Ratio Test” is used?
8. Define Sequential Probability Ratio Test.
9. Write any two applications of chi-square distribution in testing.
10. Write a short note on “Sign Test”.

 

 

PART – B

Answer any FIVE questions.                                                        5 X 8=40 marks

 

11. Let X₁, … X₁₀ be a random sample of size 10 from a normal distribution      N (0, ² ). Find  a best critical region of size  = 0.05 for testing Ho : ² = 1  against H₁: ² = 2.
12. Let X_1, X₂,…. X_n be a random sample from a distribution with the p.d.f.                         f(x;q) = q x^q-1   0 < x <  zero elsewhere, where .  Find a sufficient statistic  for q and show that a UMPT of Ho : q = 6 against H₁ : q < 6 is based on this statistic.
13. Let X₁,X₂, …, X_n be a random sample from the normal distribution N(q,1). Show that the likelihood ratio principle for testing Ho : q = q¢, where q¢ specified, against H₁ : q q¢ leads to the inequality
14. Let X have a Poisson distribution with mean q. Find the sequential probability ratio test for testing Ho : q = 0.02 against H₁ : q = 0.07.  Show that this test can be based upon the statistics   If  a₀ = 0.20 and =0.10,  find c_o (n) and    c₁ (n).
15. Let X_1,X_2,… X_n denote a random sample from a distribution that is N (q,1). where the mean q is unknown. Show that there is no uniformly most powerful test of the simple hypothesis Ho: q = q¢, Where q’ is a fixed number, against the alternative composite hypothesis H₁ : ¢.
16. Let X_1, X₂, … X_n be a random sample from a Bernoulli distribution with parameter p, where 0<p< 1. Show that the distribution has a  monotone likelihood  ratio in the statistic Y = .

 

 

 

 

 

 

 

17. The demand for a particular spare part in a factory was found to vary from day-to-day.  In a sample study the following information was obtained.

Day	Mon	Tue	Wed	Thu	Fri	Sat
No of parts demanded	1124	1125	1110	1120	1126	1115

 

Using chi-square test, test the hypothesis that the number of parts demanded does not depend on the day of  the week. Use 1% significance level.

 

18. The lengths in centimeters of n = 9 fish of a particular species captured off the

New England coast were 32.5, 27.6, 29.3, 30.1, 15.5, 21.7, 22.8, 21.2, 19.0.

Use Wilcoxon test to test Ho : m = 3.7 against the alternative hypothesis

H₁ : m > 3.7 at 5% significance level.

 

PART – C

Answer any TWO questions.                                                     2 X 20 =40 marks

 

19. (a) State and prove Neyman –Pearson theorem.
    • Consider a distribution having a p.d.f. of the form f (x ; q)  = q^x (1-q)^1-x, x = 0, 1

= 0, otherwise.

Let Ho : q = and  H₁: q >. Use the central limit theorem to determine the sample size n of a random sample so that a uniformly most powerful test of H₀  against H₁ has a power function K(q), with approximately K = 0.05 and K = 0.90.                  (10+10)

20. Let the independent random variables X and Y have distributions that are N(q_1,q₃) and N (q_2, q₃) respectively,  where the means q_1, q₂ and common variance q₃ are unknown.  If X ₁, X₂, ….,X_n and Y₁, Y_{2, …} Y_m are independent random samples from these distributions, derive a likelihood ratio test for testing H₀ : q₁=q_{2 ,}  unspecified and q₃ unspecified against all alternatives.
21. (a) Let X be N (0,q) and let q’ = 4, q” = 9. a₀ = 0.05 and =0.10. Show that

the sequential probability ratio test can be based on the statistic

Determine  c₀ (n) and c₁ (n).

 

 

 

 

(b)  A cigarette manufacturing firm claims that its brand A of the cigarettes

outsells its brand B by 8%. If it is found that 42 out of a sample of 200

smokers  prefer brand A and 18 out of another random sample of 100

smokers prefer brand B  test whether the 8% difference is a valid claim.  Use

5% level of significance.                                                                     (10 + 10)

22. (a) Below is given the distribution of hair colours for either sex in a University.

Hair colour

		Fair	Red	Medium	Dark	Jet black
Sex	Boys	592	119	849	504	36
	Girls	544	97	677	451	14

Test the homogeneity of hair colour for either sex.  Use 5% significance level.

• Using run test, test for randomness for the following data :

15  77  01  65  69  58  40  81  16  16  20  00  84  22  28  26  46  66  36  86  66  17  43  49  85  40  51  40  10 .                                                                                                    (10+10)

 

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