# ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern): Questions 14 - 18 of 39

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## Question number: 14

» Estimation » Estimation Methods » Maximum Likelihood

Appeared in Year: 2014

### Describe in Detail

x_{1 }, x _{2 }, …, x _{n } be a random sample from the following distribution

### Explanation

Let x _{1 }, x _{2 }, …, x _{n } be a random sample from f (x, α) and let L (α| x) denote the likelihood function. Then

The log-likelihood function is

We do not differentiable log L with respect to α because this is free from

## Question number: 15

» 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: 16

» Linear Models » Use of G-Inverse

Appeared in Year: 2014

### Describe in Detail

Obtain a g-inverse of A given below and verify that AA ^{-} A = A

### Explanation

Let A be a full rank m n matrix. By full rank we mean rank (A) = min {m, n}.

If m < n, then A has a right inverse given by

If m > n, then A has a left inverse given by

Our matrix A is

## Question number: 17

» 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.

## Question number: 18

» 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