# 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

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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

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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

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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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