LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2003

ST – 5501/STA506 – TESTING OF HYPOTHESIS

05.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL questions.                                                                                   (10×2=20 marks)

 

1. Define the Best critical region for testing a simple hypothesis against an alternative simple hypothesis.
2. When do you say that a family of probability density functions has a monotone likelihood ratio property?
3. Explain likelihood ratio criterion.
4. What is the test statistic used for testing the significance of correlation in a bivariate normal population? What is the distribution of the statistic?
5. A coin is tossed 100 times. If, the number of heads obtained is 45, can you say the coin is unbiased? justify.
6. What are non-parametric methods?
7. 400 oranges are taken from a large consignment and 40 are found to be bad. Find the limits in which the percentage of bad oranges is likely to come at 95% confidence.
8. Define Type I error and Type II error.
9. Explain p-value.
10. Arrange the following sets of x and y values. How many runs you get?

x          40        50        48        60

y          45        51        52        55

 

SECTION-B

Answer any FIVE questions.                                                                           (5×8=40 marks)

 

11. Let X_1, X₂, .. X_n be a random sample from N(0,1). Verify whether uniformly most powerful test exists for testing H_o :  q = q₁ vs H₁ : q ¹ q₁ .
12. Let x have the probability mass function

P [X = x]  =  x = 0, 1,2, ..M.,

It is decided to test H_o : M £ M₀ vs H₁ : M > M₀  based on a sample of size 1 from this distribution.  Derive the uniformly most powerful test.

13. In a locality 100 persons are randomly selected and asked about their educational attainments. The results one as under

Education

Middle            High school     College

Male                   10                       15                   25

Female                25                        10                  15

Does education depend on sex?

14. It is decided to study the model

X_i = a + b (c_{i  –} ) + e_i

with E(x_i) = a + b (c_i – ,    V(X_i) = s².

a random sample of size n = 10 yielded  = 67

= 2.1    = 288 and .

Test H₀: b = 0 vs H₁ : b ¹ 0 .  Also find 95% confidence interval for b.

15. Let X have the pdf f ( x ; q) = q^x (1-q)^1-x x = 0, 1 zero elsewhere. We test H₀: q = against H₁: q < .   Let X_1, X₂, X₃, X₄, X₅ be a random simple from this distribution. Obtain the uniformly most powerful test with a =
16. Explain Sign Test.
17. Explain how will you construct the confidence interval for ratio of variances, when there are two random samples from 2 independent normal populations. suppose two independent samples of sizes m = 15 and n = 10 yielded

 

 

Find the 98% confidence interval for s |  s.

18. It is known that the random variable X has a pdf

f (x ; q ) =

0        ,    elsewhere.

It is decided to test H₀: q =    Vs H₁: q = 4.

If a random sample of size 2 is observed, find probability of Type I error and probability of

type II error.

 

SECTION-C

Answer any TWO questions.                                                                           (2×20=40 marks)

 

19. a) State and prove Neyman – pearson theorem. (12)
20. b) Let X have the pdf f( x, q) = 0 < x < q

Let Y₁ < Y₂ < Y₃  < Y₄  denote the order statistics of a random sample of size 4 from

this distribution.

We reject H₀: q = 1 and accept H₁: q ¹ 1  if Y₄ £  or Y₄ ³ 1.  Find the Power function. (8)

20. a) Derive likelihood ration test for testify

H₀: m₁ =m₂  Vs       H₁: m₁ ¹ m₂

When the two random samples are drawn from two independent normal populations with mean m₁ and m₂ and with common unknown variance.                                                 (10)

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

Sample I          9          11        13        11        15        9          12        14

Sample II        10        12        10        14        9          8          10

Is the difference between means of samples significant. (Assume common variance)                                                                                                                                             (10)

21. a) Explain Run test for equality of distributions.                   (12)
22. b) Apply the Mann-Whitney – Wilcoxon test for the following data to test

H₀: F_x =F_y  Vs  H₁: F_x = F_y.

X         4.3       5.9       4.9       3.1       5.3       6.4       6.2       3.8       7.5       5.8

Y         5.5       7.9       6.8         9        5.6       6.3       8.5       4.6       7.1                   (8)

22. a) Explain the procedure of testing equality of two proportions. Also obtain the 95%

confidence interval for the difference in proportions.                                               (10)

1. b) Explain sequential probability ratio test           (10)

 

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