LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – STATISTICS
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)
- Define a simple and composite hypothesis.
- Distinguish between a randomized and non-randomized test.
- Provide the form of one-parameter exponential family.
- When a test of hypothesis is called uniformly most powerful?
- Define likelihood ratio test.
- What is a sequential probability ratio test?
- Write two uses of chi-square distribution in tests of significance.
- Write the 95% confidence interval for the population mean based on large sample.
- Define the concept of a quantile of a distribution of a random variable of the continuous type.
- Write a note on the sign test.
PART – B
Answer any Five questions. (5 x 8 = 40 Marks)
- If X1 X2 …..Xn 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.
- 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.
- 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.
- 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 Ho : q = q0 against H1 : q ¹ q0.
- 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).
- 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.
- 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.
- 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)
- (a) State and prove Neyman – Pearson theorem.
(b) If X1 X2 …..Xn is a random sample from a distribution having p.d.f. of the form f (x; q) = q xq–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,…..xn) : (10+10)
- Let X1 X2 …..Xn be a random sample from N (q1, 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.
- (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)
- 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 : px = py against H0 : px < py. 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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