# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 157 - 161 of 165

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## Question number: 157

» Statistical Methods » Tests of Significance » F-Test

Appeared in Year: 2014

### Describe in Detail

Explain the procedure for testing the hypothesis of equality of variances of two independent normal populations when population means are unknown. Write down the sampling distribution of the statistic. A sample of size 10 is drawn from each of two uncorrelated normal populations. Sample means and variances are:

1 ^{st} population: mean = 7, variance = 26

2 ^{nd} population: mean = 4; variance = 10

Test at 5 % level of significance whether the first population has greater standard deviation than that of the second population. [Given F _{0.05, 9, 9} = 3.18]

### Explanation

Let X _{1}, X _{2}, …, X _{n} and Y _{1}, Y _{2}, …Y _{m} are the samples taken from independent N (µ _{1}, σ _{1}^{2}) and N (µ _{2}, σ _{2}^{2}).

In this question the hypothesis for testing is

## Question number: 158

» Statistical Methods » Non-Parametric Test » Mann-Whitney

Appeared in Year: 2009

### Describe in Detail

Given the following data:

x | 0 | 1 | 3 |

f (x) | 1 | 3 | 55 |

find a polynomial P (x) of degree 2 or less so that P (x) = f (x) at the tabulated values of x. Hence

approximate f (2).

### Explanation

Let the a polynomial is P (X) = kx ^{2} +lx + m, where k, l, m are constants which is determine by using Lagrange’s interpolation polynomial because the x values is not equal interval.

where

x _{0} =0, x _{1} =1 _{, }, x _{2} =3 and f

## Question number: 159

» Statistical Methods » Tests of Significance » Chi-Square

Appeared in Year: 2009

### Describe in Detail

Explain the method of testing normality by using chi-squared test.

### Explanation

Let X is a random variable follows normal distribution with mean µ and variance σ ^{2}. The testing of null hypothesis is that the population variance σ ^{2} equals a specified value against one of the usual alternatives σ ^{2} < , σ ^{2} > ,

## Question number: 160

» Probability » Laws of Total and Compound Probability

Appeared in Year: 2011

### Describe in Detail

Verify the following identities:

(i)

(ii)

### Explanation

Let A and B are two possible events in the sample space.

(i) Additive law of probability is

{Commutative property }

We know that

This shows that

(ii) This also show by additive law of probability

Let assume BUC = D, then

Putting the

## Question number: 161

» Numerical Analysis » Interpolation Formulae » Newton-Gregory

Appeared in Year: 2010

### Describe in Detail

Estimate U _{2} from the following table:

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

U | 7 | - | 13 | 21 | 37 |

### Explanation

To find the missing value, we use binomial expansion method. Here 4 values are known, we would take fourth order finite difference zero. Thus,

Here for x = 1, U _{0} =7, U _{1} =? , U _{2} =13, U _{3} =21, U _{4} =37