BSc Sem VI 2022 2023 Apr 2023 STATISTICS TESTING OF HYPOTHESIS Question Paper - Mumbai University | munotes

Questions asked in this paper

• 2) Figures to the right indicate full marks

1. Q1 a) Define the following terms with suitable example 10 marks
   • i. Simple Hypothesis
   • v. Power of the test
   • b) State and prove Neyman-Pearson Lemma for obtaining the best test for testing simple Null hypothesis against simple alternative hypothesis 10
   • p) I) variable X has the following to test the hypothesis Ho: 0 = 4 against = 5. Determine the value of ‘c’ if the critical region is given by x >c. a=0.025. Also calculate the power of test II) If x => | is the critical region for testing Ho: 8 =2 against Hi: 8 =1 based [05] on a single observation from a population with p.d.f Calculate size and power of the test 5
   • q) Derive the Most powerful test of size a for testing Hp : 0 = against H, : 0 = 6, when a sample of size n is taken from Bernoulli (8) for the following cases 10
   • a) Define Uniformly Most Powerful Test (UMPT). Obtain UMPT of to test : 0 = against H, : 0>6,, where random sample of size n drawn from 10
   • b) I) Define Likelihood Ratio Test (LRT) and write down its procedure. Comment: Level of significance is always equal to the probability of Pp) Obtain UMPT of size to test Hy : 0 = against H, : X,, [10] is arandom sample of size n drawn from Exponential 7
   • q) Let sample drawn from N( 07) population where is known. Derive LRT to test Hy : = against Paper Subject Code: 88619 Statistics: Testing of Hypothesis 10
2. Q3 a) Discuss the test procedure of Sequential Probability Ratio Test (SPRT) for testing a simple null hypothesis against a simple alternative Differentiate between Sequential test procedures with Neyman 10 marks
   • b) Construct SPRT of strength (a, for testing against
   • 80), when a random variable X follows Binomial distribution with
   • p) Arandom variable X follows a Poisson distribution with parameter A Derive SPRT of strength (a, B) to test the hypothesis Ho: A = Ao v/s Hi: A 10
   • q) Construct SPRT of strength (a, B) for testing Ho:u= against Hi: u= > from N(u, where is known 10
3. Q4 a) Stating the assumption clearly, explain the Sign test for single sample. Also explain the Normal approximation for sign test 12 marks
   • b) Explain the median test for two independent samples. 8
   • p) Explain Wilcoxon matched pair signed rank test for two related II) Explain the Wald-Wolfowitz Run test 5
   • q) Explain Kruskal Wallis one way ANOVA by Ranks test stating all the 10
4. Q5 Attempt Any Two sub questions
   • a) Define the following terms with suitable example 10
   • 1. Alternative hypothesis
   • 3. P-value
   • 5. Acceptance region
   • b) Let be arandom sample drawn from N( population where p is known. Derive LRT to test Hp : o? against 10
   • c) variable X follows an Exponential distribution with unknown parameter Derive SPRT of strength (a, for testing Ho : 10
   • d) Explain Friedman Two way ANOVA by ranks test stating the 10

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