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

Get 1 year subscription: Access detailed explanations (illustrated with images and videos) to **39** questions. Access all new questions we will add tracking exam-pattern and syllabus changes. View Sample Explanation or View Features.

Rs. 200.00 or

## Question number: 1

» Hypotheses Testing » Likelihood Ratio Test » ASN Function

Appeared in Year: 2015

### 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. ; i = 1, 2, …, k. We have to test

In the __X__ population the sample is = { (x _{i1}, x _{i2}, …, x _{ini})

The

## Question number: 2

» Hypotheses Testing » Hypothesis » Composite

Appeared in Year: 2014

### Describe in Detail

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

Where z _{α/2 } is the critical value = 1.96

σ is the standard deviation = 2

E is the margin of error=

## Question number: 3

» Hypotheses Testing » Randomised Test

Appeared in Year: 2014

### 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

## Question number: 4

» Hypotheses Testing » Likelihood Ratio Test » ASN Function

Appeared in Year: 2014

### 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

Under H _{0 }, the parameter space is

The likelihood function is

Under the whole space, the unrestricted MLE is

Under H _{0 }, the restricted MLE is

The statistic is

The likelihood

## Question number: 5

» Hypotheses Testing » Two Kinds of Error

Appeared in Year: 2014

### Describe in Detail

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

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.