Hypotheses Testing (ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern)): Questions 1 - 5 of 9

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Question number: 1

» Hypotheses Testing » Likelihood Ratio Test » ASN Function

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Derive the likelihood ratio test for comparing the means of k independent homoscedastic normal populations.

Explanation

Given that there are k independent homoscedastic normal populations that is the variance is same i. e. Equation ; i = 1, 2, …, k. We have to test

Equation

In the X population the sample is = { (x i1, x i2, …, x ini)

The

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Question number: 2

» Hypotheses Testing » Hypothesis » Composite

Appeared in Year: 2014

Essay Question▾

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A sample of size n from normal distribution N (θ, σ 2 ) with σ 2 =4 was observed. 95 % confidence interval for θ was computed from the above sample. Find the value of n if the confidence interval is (9.02, 10.98).

Explanation

The Margin of error is defined as

Equation

Where z α/2 is the critical value = 1.96

σ is the standard deviation = 2

E is the margin of error= Equation

Equation

Equation

Question number: 3

» Hypotheses Testing » Randomised Test

Appeared in Year: 2014

Essay Question▾

Describe in Detail

Bring out the difference between a randomized and non-randomized test. Explain how the decision based on a randomized test can be taken in the discrete set-up.

Explanation

A test T of a hypothesis H is said to be non-randomized test if the decision about the rejection or acceptance of H is based on a test statistic. H is rejected if the test statistic lies in the critical region otherwise accepted. A randomized test is one in which

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Question number: 4

» Hypotheses Testing » Likelihood Ratio Test » ASN Function

Appeared in Year: 2014

Essay Question▾

Describe in Detail

x 1 , x 2 , …, x n is a random sample from N (θ, σ 2 ) (σ 2 not specified). Derive likelihood ratio test of testing H 0 : θ=θ 0 against H 1 : θ ≠ θ 0 .

Explanation

Under the given model, the parameter space is

Equation

Under H 0 , the parameter space is

Equation

The likelihood function is

Equation

Under the whole space, the unrestricted MLE is

Equation

Under H 0 , the restricted MLE is

Equation

The statistic is

Equation

Equation

Equation

Equation

The likelihood

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Question number: 5

» Hypotheses Testing » Two Kinds of Error

Appeared in Year: 2014

Essay Question▾

Describe in Detail

A single observation of a r. v. having a geometric distribution with pmf

Equation

The null hypothesis is H 0 : θ=0.5 against the alternative hypothesis H 1 : θ=0.6 is rejected if the observed value of r. v. is greater than equal to 5. Find the probabilities of type I error and type II error.

Explanation

The type I error is the probability that reject H 0 when it is true denote by α and type II error is the probability that accept H 0 when it is false denote by β

Given that the null hypothesis is rejected if the observed value of r. v.

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