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

 

 

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

 

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

 

SECTION – B

 

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

 

11. 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.
12. Obtain the Best Critical Region for testing H: q = q₁ versus K: q = q₂ ( > q­₁) when a sample of size ‘n’ is drawn from f(x; q ) = , 0 < x < 1 ( q > 0)
13. Show that the family of Binomial densities has MLR property.
14. Explain SPRT in detail.
15. Derive the likelihood ratio test for testing against based on a sample of size 10 drawn from
16. Explain the process of testing the significance of correlation coefficient.
17. Obtain the confidence interval for the mean of a normal distribution with unknown variance.
18. Explain in detail Median test.

(P.T.O)

SECTION – C

 

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

 

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

of size two drawn from

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

 

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

drawn from

 

21. 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.

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

with common unknown variance.

 

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

 

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – APRIL 2011

ST 5505 – TESTING OF HYPOTHESES

 

 

 

Date : 18-04-2011              Dept. No.                                                    Max. : 100 Marks

Time : 9:00 – 12:00

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – STATISTICSFIFTH SEMESTER – APRIL 2012ST 5505/ST 5501 – TESTING OF HYPOTHESES
Date : 27-04-2012 Dept. No.         Max. : 100 Marks                 Time : 9:00 – 12:00
PART – A
Answer ALL questions: (10×2=20 Marks)
1. Distinguish between simple and composite hypothesis.2. What is meant by testing of hypothesis? 3. Define randomized test.4. Explain the meaning of level of significance. What does 5% level of significance imply?5. Which tests of hypothesis are called two-tailed tests? Give an example for it. 6. State the Likelihood Ratio Criterion.7. Write down the steps involved in a test of significance procedure for large samples.8. Define non-randomized test.9. Which types of tests are called non-parametric tests?10. Mention any two advantages of non-parametric tests.
PART – B
Answer any FIVE questions: (5×8=40 Marks)
11. Let  be the probability that a coin will fall head in a single toss in order to test  against . The coin is tossed 5 times and  is rejected if more than 3 heads are obtained. Find the probability of Type I error and power of the test.  12. Let   be a random sample from , where   is known. Find a UMP test for testing   against .13. Derive the likelihood ratio test for the mean of a normal population  when   is known.14. Derive the likelihood ratio test for the variance of a normal population  when   is known.15. Describe likelihood ratio test procedure and state its properties. 16. What is paired t test? What are its assumptions? Explain the test procedure.17. Explain the test procedure for testing the randomness of a sample.  18. Discuss the procedure for median tests.PART – C
Answer any TWO questions:       (2×20=40 Marks)
19. (a) State and prove Neymann-Pearson Lemma.      (b) Illustrate that UMP test does not exist always.20. (a) What are the applications of chi-square distribution in testing of hypothesis.             (b) Explain the test procedure for testing equality of variances of two normal populations.      21. (a) What are the applications of t-distribution in testing of hypothesis?      (b) Explain Wald-Wolfowitz Run test for two samples.22. (a) Explain the Chi-square test of independence of attributes in contingency table.             (b) Explain the sign test for one sample.

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2012

ST 5505/ST 5501 – TESTING OF HYPOTHESES

 

 

 

Date : 03/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

 

Answer ALL Questions:                                                                                           ( 10 x 2 = 20 Marks )

 

1. Distinguish between Simple and Composite hypotheses.
2. Define Best Critical Region.
3. Define Exponential Distribution.
4. When do you call a test uniformly most powerful?
5. Define SPRT for testing Ho against H₁.
6. State the ASN function for the SPRT for testing H_o: q = q₀ against H₁: q = q₁.
7. What do you mean by one-tailed and two-tailed tests?
8. State the assumptions for Student’s t-test.
9. Mention the assumptions associated with Non-parametric tests.
10. State the situations where Sign test can be applied.

 

PART – B

Answer any FIVE  questions:                                                                                  ( 5 x 8 = 40 Marks )

 

11. Explain the concept of critical region.

12 Define and elaborate two types of errors in testing of hypothesis.

13. Discuss the general approach of likelihood ratio test.
14. Find the LRT of H_o: q = q₀ against H₁: q ≠ q_o based on sample of size 1 from the density

 

f ( x, q ) = 2 ( q – x ) / q²  ,  0 < x < q

15. Explain the concepts
16. i) Level of Significance
17. ii) Null and Alternative hypotheses.
18. A manufacturer of dry cells claimed that the life of their cells is 24.0 hours. A sample of

10 cells had mean life of 22.5 hours with a standard deviation of 3.0 hours.  On the basis of

available information, test whether the claim of the manufacturer is correct.

17  In a breeding experiment, the ratio of off-spring in four classes was expected to be 1:3:3:9.

The experiment yielded the data as follows:

Classes                            AA           Aa          aA             aa

No.of offsprings:               8              29          37           102

 

Test whether the given data is in agreement with the hypothetical ratio.

18. Use the sign test to see if there is a difference between the number of days required to collect

an account receivable before and after a new collection policy. Use the 00.5 significance level

Before:  33  36  41  32  39  47  34  29  32  34  40  42  33  36  27

After  :  35  29  38  34   37  47  36  32  30  34  41  38  37  35  28

 

PART – C

Answer any TWO questions:                                                                                   (2 x 20 = 40 Marks )

 

19 a) State and Prove Neymann-Pearson Lemma.

 

1. b) A sample of size 1 is taken from density

f ( x, q ) = 2 ( q – x ) / q²  ,  0 < x < q

= 0   else where

Find an Most Powerful test of H_o: q = q₀ versus H₁: q = q₁ ;  q₀ > q₁  at level α .

 

20 a) Describe the sequential procedure for testing H_o: q = q₀ against H₁: q ≠ q₁ where q is the

parameter  of the Poisson distribution.

 

63. b) The heights of ten children selected at random from a given locality had a mean 63.2 cms

and variance 6.25 cms.  Test at 5 % level of significance the hypothesis that the children of

the given locality are on the average less than 65 cms in all. Given for 9 degrees of freedom

P( t.> 1.83) = 0.05.

 

21. a) Explain Chi-square test of Goodness of fit.

 

1.  b)  The following table gives the number of aircraft accidents that occurred during the seven

days of the week.  Find whether the accidents are uniformly distributed over the week.

Days                   :  Mon     Tue    Wed    Thur    Fri     Sat    Total

No.of accidents  :    14        18       12        11      15      14      84

 

22. a) Find 99 % confidence limits for the parameter l in Poisson distribution.

 

1. b) Apply Median Test for the following data:

 

X:     27   31   32   33   34   29   35

Y:     28   30   30   24   25   26

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2012

ST 5505/ST 5501 – TESTING OF HYPOTHESES

 

 

 

Date : 03/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

 

Answer ALL questions:                      (10 x 2 = 20 marks)

 

1. Distinguish between Simple and Composite hypotheses.
2. Define Best Critical Region.
3. Define Exponential Distribution.
4. When do you call a test uniformly most powerful?
5. Define SPRT for testing H_O against H₁.
6. State the ASN function for the SPRT for testing H_o: q = q₀ against H₁: q = q₁.
7. What do you mean by one-tailed and two-tailed tests?
8. State the assumptions for Student’s t-test.
9. Mention the assumptions associat

 

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