LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

** ****B.Sc.** DEGREE EXAMINATION – **STATISTICS**

FIFTH SEMESTER – NOVEMBER 2010

# ST 5505/ST 5501 – TESTING OF HYPOTHESES

Date : 01-11-10 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

__SECTION – A__

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**ANSWER ALL QUESTIONS. (10 X 2 =20 marks)**

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- What is a composite hypothesis? Give an example.
- Define: Critical region.
- Given an example of a density function which is not a member of the one parameter exponential family.
- When do you say a given family of density functions has MLR property?
- What are Type I and II errors?
- Define: Likelihood ratio.
- What are confidence intervals?
- State the test statistic for testing the equality of variances of two normal populations.
- Define: Empirical Distribution Function.
- Mention the use of Kolmogrov one sample test.

__SECTION – B__

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**ANSWER ANY FIVE QUESTIONS (5 X 8 =40 marks)**

- A sample of size one is drawn from a population with probability density function . To test the hypothesis against the following test is suggested: “Reject H if x > 4”. Compute the size and power of the test.
- Obtain the Best Critical Region for testing H:
*q =**q*versus_{1}*K:**q =**q*( > q_{2}_{1}) when a sample of size ‘n’ is drawn from f(*x*;*q*) = , 0 <*x*< 1 (*q*> 0) - Show that the family of Binomial densities has MLR property.
- Explain SPRT in detail.
- Derive the likelihood ratio test for testing against based on a sample of size 10 drawn from
- Explain the process of testing the significance of correlation coefficient.
- Obtain the confidence interval for the mean of a normal distribution with unknown variance.
- Explain in detail Median test.

(P.T.O)

__SECTION – C__

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**ANSWER ANY TWO QUESTIONS (2 X 20 = 40 marks)**

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- a) State and prove Neyman Pearson lemma.
- b) Derive the MPT of level 0.05 for testing against based on a sample

of size two drawn from

- a) Show that the family of Uniform distributions has MLR property.

- b) Derive the UMPT of level 0.05 for testing against based on a sample of size 10

drawn from

- a) Obtain the SPRT for testing H: p =1/2 versus K: p = 1/3 when a sample is drawn sequentially

from B(1,p) with α = β = 0.1.

- b) Explain the procedure for testing the equality of means of two independent normal populations

with common unknown variance.

- a) Explain Mann-Whitney U test.
- b) Write a descriptive note on non-parametric methods.