# Hypotheses Testing (ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern)): Questions 1 - 6 of 9

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## 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 parameter space is

The parameter space under H _{0} is

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## 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=

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## 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 no test statistics is involved. The decision about H is taken on the basis of some p

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## 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 stati

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## 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. is greater than equal to 5 that means X ≥ 5.

For type I error,

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## Question number: 6

» Hypotheses Testing » Randomised Test

Appeared in Year: 2014

### Describe in Detail

chart is used to control the mean of normally distributed characteristic. It is know that . The center line is 200. If the process mean shifts to 188, find the probability that this shift is detected on the first subsequent sample.

### Explanation

Type II error is the probability of saying that the process is under control when it is not under control, that is, the probability of a point falling within control limits after the shift the mean. This is also known as the probability of not detecting the shift after shift the mean. Thus,

The probability of detecting th

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