ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern): Questions 13 - 18 of 39

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Question number: 13

» Statistical Quality Control » Control Charts » Attributes

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Appeared in Year: 2014

Essay Question▾

Describe in Detail

Find 3-sigma control limits for a

(i) C-chart with process average equal to 4 non-conformities

(ii) U chart with process average c = 4, and n = 4.

Explanation

the 3-sigma control limits is

(i) Given that

Control limits for the C chart with a known non-conformities is

(ii) Given that c = 4 and n

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Question number: 14

» Estimation » Estimation Methods » Maximum Likelihood

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Appeared in Year: 2014

Essay Question▾

Describe in Detail

x1 , 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

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Question number: 15

» Hypotheses Testing » Likelihood Ratio Test » ASN Function

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Appeared in Year: 2014

Essay Question▾

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

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Question number: 16

» Linear Models » Use of G-Inverse

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Appeared in Year: 2014

Essay Question▾

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 2 × 3 with rank two, so

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Question number: 17

» Hypotheses Testing » Two Kinds of Error

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Appeared in Year: 2014

Essay Question▾

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: 18

» Hypotheses Testing » Randomised Test

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Appeared in Year: 2014

Essay Question▾

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,

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