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 H0 and H1 given below. Find M. P. test with α = 0.03
x | 1 | 2 | 3 | 4 | 5 | 6 |
f0 (x) | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.95 |
f1 (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) | ||||||
FZ (z) | 1 |
The next step is to find k > 0 and γ such that
Consider k such that Fz (k-) ⩽ α ⩽ Fz (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 y1 , y2, … , yn 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 tα/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 = (X1, … , Xk) and also let Ek be the set of all points in k dimensional Euclidean space Rk, for which we reject H0 using the sequential probability ratio test. Also, let Fk be the set of all points in Rk for which we accept H0. Notice that (Ek, k = 1,2, …) are mutually disjoint and (Fk, k = 1,2, …) are also mutually disjoint. Assume that Th…
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