# ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern): Questions 19 - 24 of 39

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## Question 19

Appeared in Year: *2014*

### Describe in Detail

Essay▾Let X be r. v. with pmf under H_{0} and H_{1} given below. Find M. P. test with α = 0.03

x | 1 | 2 | 3 | 4 | 5 | 6 |

f_{0} (x) | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.95 |

f_{1} (x) | 0.05 | 0.04 | 0.03 | 0.02 | 0.01 | 0.85 |

### Explanation

To find the M. P. test, first we compute

Z | 5 | 4 | 3 | 2 | 1 | 0.8947 |

The possible value of Z is

Z | 5 | 4 | 3 | 2 | 1 | 0.8947 |

P (Z = z) | ||||||

F_{Z} (z) | 1 |

The next step is to find k > 0 and γ such that

Consider k such that F_{z} (k-) ⩽ α ⩽ F_{z} (k)

So, we choose k = 5,

The M. P. test is

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## Question 20

Appeared in Year: *2014*

### Describe in Detail

Essay▾Let X be exponentially distributed with parameter θ. Obtain MLE of θ based on a sample of size n, from the above distribution.

### Explanation

Let X be exponentially distributed with parameter θ.

The likelihood function is

The log-likelihood function is

Differentiable with respect to θ, equating to zero

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## Question 21

Appeared in Year: *2015*

### Describe in Detail

Essay▾Let y_{1} , y_{2, … ,} y_{n} be a random sample from N (µ, σ ^{2}) where µ and σ ^{2} are both unknown. Obtain a confidence interval of µ with confidence coefficient (1-α)

### Explanation

When population mean and population standard deviation in not know. If is the samplemean and replace σ by its estimate s and t_{α/2} be the critical value of the student t-test such that have of the area on the left hand side and other half on the right side that is

For first inequality we get

From second

Combining the inequalities

The confidence interva…

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## Question 22

Appeared in Year: *2015*

### Describe in Detail

Essay▾Obtain the sufficient statistics for the following distribution.

(i)

(ii)

### Explanation

By using factorization theorem, the condition is that

where h (x) is free from θ and g (.) depends on __X__ only through T.

(i)

The joint pdf of random sample is

Let T = . By factorization theorem

h (x) = 1,

So, h (x) is free from θ and g depends on sample only through T = .

Thus, T = is sufficient for this distribution.

(ii)

The joint pdf of random sample …

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## Question 23

Appeared in Year: *2015*

### Describe in Detail

Essay▾Show that T ^{2} statistic is invariant under changes in the unit of measurements for a p×1 random vector X of the form __Y__ = C __X__ + __d__ where C is a p×p nonsingular matrix, __d__ is a p×1 vector.

### Explanation

T ^{2} statistic can be computed from __X__ . Here we have to compute it form __Y__ and both will be same. Here

Let , then and

So, T ^{2} -statistic comes into existence for testing hypothesis of the form

Since is known vector,

If we have a setup

and

T ^{2} computed from __X__ is

This is the T ^{2} statistic which is computed from __Y__ vector.

## Question 24

Appeared in Year: *2015*

### Describe in Detail

Essay▾For a sequential probability ratio test of strength (α, β) and stopping bounds are A and B (B < A) , show that A⩽1-β/α and B ⩾ β/1-α

### Explanation

Let X = (X_{1}, … , X_{k}) and also let E_{k} be the set of all points in k dimensional Euclidean space R_{k}, for which we reject H_{0} using the sequential probability ratio test. Also, let F_{k} be the set of all points in R_{k} for which we accept H_{0}. Notice that (E_{k}, k = 1,2, …) are mutually disjoint and (F_{k}, k = 1,2, …) are also mutually disjoint. Assume that Th…

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