LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AC 19

FIFTH SEMESTER – APRIL 2006

                                                    ST 5501 – TESTING OF HYPOTHESIS

(Also equivalent to STA 506)

 

 

Date & Time : 22-04-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

PART – A

Answer all the questions.                                               (10 x 2 = 20 Marks)

 

1. Define a simple and composite hypothesis.
2. Distinguish between a randomized and non-randomized test.
3. Provide the form of one-parameter exponential family.
4. When a test of hypothesis is called uniformly most powerful?
5. Define likelihood ratio test.
6. What is a sequential probability ratio test?
7. Write two uses of chi-square distribution in tests of significance.
8. Write the 95% confidence interval for the population mean based on large sample.
9. Define the concept of a quantile of a distribution of a random variable of the continuous type.
10. Write a note on the sign test.

 

PART – B

Answer any Five questions.                                                       (5 x 8 = 40 Marks)

11. If X₁ X₂ …..X_n is a random sample from N (q, 1) ,  q Î R , find a most powerful test for testing H₀ : q = 0 against H₁ : q = 1.
12. Let X₁ 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.  For this test determine the significance level   a and the power at  q = 0.5.
13. X₁ X₂ …..X₂₅ denote a random sample of size 25 from a normal distribution

N (q, 100).  Find a uniformly most powerful critical region of size a = 0.10 for testing H₀: q  = 75 against H₁: q > 75.

 

14. Let Y₁ < Y₂ < …. < Y₅ be the order statistics of a random sample of size n =5 from a distribution with p.d.f

f (x; q) = ½  e^{– |x– q|},   x Î R, for all real q.  Find the likelihood ratio test for testing H_o : q = q₀ against H₁ : q ¹ q_0.

 

15. Let X be N (0, q) and let q¢ = 4, q¢¢ = 9, a₀ = 0.05 and b₀ = 0.10. Show that the SPRT can be based upon the statistic .  Determine c₀ (n) and c₁ (n).

 

16. A random sample of 100 units was drawn from a large population. The mean and standard error of the sample are respectively 45 and 8.   Obtain 95% confidence interval for the population mean.

 

17. For the following data:

X₁  :        25        30        45        52        65        75        80        42        50        60

X₂ :         60        40        35        50        60        72        63        40        55        62

Test whether the two population variances are equal.  Use 5% significance level.

 

18. Let X denote the length of time in seconds between two calls entering a college switch board. Let p denote the unique median of this continuous type distribution.  Test H₀: p = 6.2 against H₁: p < 6.2 based on a random sample of size 20.

 

6.8          5.7       6.9       5.3       4.1       9.8       6.7       7.0       2.1       19.0     18.9

16.9        10.4     44.1     2.9       2.4       4.8       18.9     4.8       7.9

 

PART – C

Answer any Two questions.                                           (2 x 20 = 40 Marks)

 

19. (a) State and prove Neyman – Pearson theorem.

(b) If  X₁ X₂ …..X_n  is a random sample from a distribution having p.d.f. of the form f (x; q) = q x^q–1,  0 < x < 1, zero elsewhere, show that a best critical region for testing Ho: q = 1 against H₁: q = 2 is C = {(x₁, x₂,…..x_n)  :                                     (10+10)

 

20. Let X₁ X₂ …..X_n be a random sample from N (q_1, q₂) and let
    W = {(q₁, q₂) : –¥ < q₁ < ¥, 0 < q₂ < ¥} and w = {(q₁, q₂) : q₁ = 0, 0 < q₂ < ¥)}.

Perform the likelihood ratio test for testing H₀: q₁ = 0, q₂ > 0            against H₁: q₁ ¹ 0,
q₂ > 0.

21. (a) Consider the following data:

X:        32        40        50        62        70        35        42        45        19        30

Y:           30        43        55        60        65        23        32        40        20        35

Test whether the two population means are equal at 1% significance level.

 

• The number of transactions in a teller’s counter during the days of the week is given below:

Mon.             Tue.                 Wed.               Thu.                 Fri.                   Sat

33                 28                    24                    22                    30                    19

Test whether the number of transactions are uniformly distributed or not.  Use 1% level of significance.                                                                                                                    (10+10)

 

22. Let X and Y denote the weights of ground cinnamon in 120 gram tins packaged by companies A and B respectively. Use Wilcoxon test to test the hypothesis
    H₀ : p_x = p_y against H₀ : p_x < p_y.  The weights of n₁ = 8 and n₂ = 8 tins of cinnamon packages by the companies A and B selected at random yielded the following observations of X and Y respectively.

X:  117.1       121.3        127.8        121.9          117.4       124.5      119.5        115.1

Y:  123.5       125.3        126.5        127.9          122.1       125.6      129.8        117.2

 

 

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