LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – STATISTICS
FIFTH SEMESTER – APRIL 2007
ST 5501 – TESTING OF HYPOTHESIS
Date & Time: 28/04/2007 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
PART – A
Answer ALL the questions. 10 X 2=20 marks
- Define a simple and composite statistical hypothesis and give an example each.
- What are Type-I and Type-II errors in testing of hypothesis. Also define the power function.
- Define a best critical region (BCR) of size for testing the simple hypothesis against an alternative simple hypothesis.
- When do you say that a BCR is uniform?
- When do you say that a distribution belongs to an exponential family?
- Under what situation likelihood ratio test is used?
- Define Sequential Probability Ratio Test (SPRT).
- State any two differences between SPRT and other test procedures.
- Write the statistic for testing the equality of means when the sample is small.
- What is a non-parametric test?
PART – B
Answer any FIVE questions. 5 X 8 =40 marks
- Let Y have a binomial distribution with parameters n and p. We reject Ho: p = ½ and accept H1 = p> ½ if y Find n and c to give a power function K(p) which is such that K = 0.10 and K = 0.95 approximately.
- State and prove Neyman Pearson theorem.
- If X1, X2,…Xn is a random sample from normal distribution with mean q and variance 1, find a BCR of size a for testing Ho: q = 0 against H1 : q =1.
- Let X1, X2, … Xn denote a random sample from a distribution that is N(q, 1). Show that there is no uniformly most powerful test of the simple hypothesis Ho: q = q¢ where q¢ is a fixed number , against the alternative composite hypothesis H1 : qq¢.
- Let X1, X2, ….Xn be a random sample from a Poisson distribution with parameter q where q>0. Show that the distribution has a monotone likelihood ratio in the statistic Y= .
- Let Y1,< Y2 < …. < Y5 be the order statistics of a random sample of size n=5 from a distribution with pdf f(x;q) = e–|x – q | , -<x<, for all real q. Find the likelihood ratio test for resting Ho: q = q0 against H1 : q q0.
- Ten individuals were chosen at random from a population and their heights were found to be in inches 63, 63, 66, 67, 68, 69, 70, 70, 71, 72. Test the hypothesis that the height in the population is 66 inches. Use 5% significance level.
- Explain the wilcoxon
PART – C
Answer any TWO questions. 2 X 20 =40 marks
- (a) 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. Determine the significance level a and the power at q = 0.5
for this test.
- Let X have a pdf of the form f (x;q) = , 0 < x < q zero elsewhere. 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 :q1 if either
y4 or Y4 Find the power function K (q), 0 < q, of the test. (10+10)
- Let the random variable X be N(q1,q2). Derive a likelihood radio test for testing H0 : q1 = 0, q2 >0 against H1 : q1 0 , q2 >0.
- (a) Let X be N(0,q) and let q¢ = 4 , q” = 9, a0 = 0.05, and o = 0.10. Show
that the sequential probability ratio test can be based upon the statistic
. Determine c0(n) and c1(n) .
- In a survey of 200 boys, of which 75 were intelligent, 40 has skilled
fathers while 85 of the intelligent boys has unskilled fathers. Do these
figures support the hypothesis that skilled fathers have intelligent boy? Use
5% significance level. (10+10)
- (a) An IQ test was administered to 5 persons before and after they were trained.
The results are given below :
|IQ before training||110||120||123||132||125|
|IQ after training||120||118||125||136||121|
Test whether there is any improvement in IQ after the training programme. Use 1%
- Let m be the median lung capacity in litres for a male freshman. Use sign test to test at the a = 0.0768 significance level, the null hypothesis Ho : m = 4.7 against the two sided alternative hypothesis H1 :.m > 4.7. The observations are : 7.6 4 4.3 5.0 5.7 6.2 4.8 4.7 5.6 5.2 3.7 4.0 5.6 6.8 4.9 3.8 5.6 (10+10)
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