LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AC 17

FIFTH SEMESTER – APRIL 2007

ST 5501 – TESTING OF HYPOTHESIS

 

 

 

Date & Time: 28/04/2007 / 1:00 – 4:00 Dept. No.                                              Max. : 100 Marks

 

 

PART – A

Answer ALL the questions.                                                        10 X 2=20 marks

1. Define a simple and composite statistical hypothesis and give an example each.
2. What are Type-I and Type-II errors in testing of hypothesis.  Also define the power function.
3. Define a best critical region (BCR) of size for testing the simple hypothesis against an alternative simple hypothesis.
4. When do you say that a BCR is uniform?
5. When do you say that a distribution belongs to an exponential family?
6. Under what situation likelihood ratio test is used?
7. Define Sequential Probability Ratio Test (SPRT).
8. State any two differences between SPRT and other test procedures.
9. Write the statistic for testing the equality of means when the sample is small.
10. What is a non-parametric test?

 

PART – B

Answer any FIVE questions.                                                        5 X 8 =40 marks

11. Let Y have a binomial distribution with parameters n and p. We reject Ho: p = ½ and accept     H₁ = p> ½ if y  Find n and c to give a power function K(p)  which is such that K  = 0.10 and K = 0.95 approximately.
12. State and prove Neyman Pearson theorem.

 

13. If X₁, X₂,…X_n is a random sample from normal distribution with mean q and variance 1, find a BCR of size a for testing Ho: q = 0 against H₁ : q =1.

 

14. Let X₁, X₂, … X_n denote a random sample from a distribution that is N(q, 1). 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 : qq¢.
15. Let X₁, X₂, ….X_n be a random sample from a Poisson distribution with parameter q where q>0. Show that the distribution has a monotone likelihood ratio in the statistic Y= .
16. Let Y₁,< Y₂ < …. < Y₅ be the order statistics of a random sample of size n=5 from a distribution with pdf f(x;q) =   e^{–|x – q |} , -<x<, for all real q.  Find the likelihood ratio test for resting Ho: q = q₀ against H₁ : q q₀.
17. Ten individuals were chosen at random from a population and their heights     were   found to be in inches 63, 63, 66, 67, 68, 69, 70, 70, 71, 72.  Test the hypothesis   that the height in the population is 66 inches. Use 5%  significance level.

 

18. Explain the wilcoxon

 

 

PART – C

Answer any TWO questions.                                                              2 X 20 =40 marks

 

19. (a) Let X_1, X₂,…, X₁₀ denote a random sample of size 10 from a Poisson

distribution with mean q.  Show that the critical region C defined by

is a best critical region for testing H₀ : q = 0.1 against

H₁ : q = 0.5.  Determine the significance level a and the power at q = 0.5

for this test.

• Let X have a pdf of the form f (x;q) = , 0 < x < q  zero elsewhere.  Let

Y₁< Y₂ < Y_{3 ,}< Y₄ denote the order statistics of a random sample of size 4

from this distribution. we reject H₀ : q = 1 and accept H₁ :q1 if either

y₄ or Y₄  Find the power function K (q),  0 < q, of the test.   (10+10)

20. Let the random variable X be N(q₁,q₂). Derive a likelihood radio test for testing                         H₀ : q₁ = 0, q₂ >0 against H₁ : q₁ 0  , q₂ >0.

 

21. (a) Let X be N(0,q) and let q¢ = 4 , q” = 9, a₀ = 0.05, and _o = 0.10.  Show

that  the sequential probability ratio test can be based upon the statistic

. Determine c₀(n) and c₁(n) .

• In a survey of 200 boys, of which 75 were intelligent, 40 has skilled

fathers while 85 of the intelligent boys has unskilled fathers.  Do these

figures support the hypothesis that skilled fathers have intelligent boy?  Use

5% significance level.                                                                       (10+10)

 

22. (a) An IQ test was administered to 5 persons before and after they were trained.

The results are given below :

 

Candidates	I	II	III	IV	V
IQ before training	110	120	123	132	125
IQ after training	120	118	125	136	121

 

Test whether there is any improvement in IQ after the training programme. Use 1%

significance level.

• Let m be the median lung capacity in litres for a male freshman. Use sign test to test at the a = 0.0768  significance level, the null hypothesis  Ho : m = 4.7 against the two sided alternative hypothesis H₁ :.m > 4.7.  The observations are : 7.6   4   4.3   5.0   5.7   6.2   4.8   4.7   5.6   5.2   3.7   4.0   5.6   6.8   4.9   3.8   5.6                                                   (10+10)

 

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