# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 157 - 162 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

Let s _{1}^{2} and s _{2}^{2} be the estimates variances of σ _{1}^{2} and σ _{2}^{2} based on sample sizes n and m.

and

Then

… (211 more words) …

## 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 (x _{0}) =1, f (x _{1}) =3, f

… (112 more words) …

## 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} > , or σ ^{2}. The appropriate statistic on which to base our decision is the chi-squared statistic.

In chi

… (131 more words) …

## 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

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## 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

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## Question number: 162

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2011

### Describe in Detail

Show that is a probability density function for an appropriate value of a. Upto what order do the moments of this p. d. f. exist?

### Explanation

To find the value of a, we known that the probability density function under the range of x is equal to one.

To find order which is exist for this p. d. f. is

Upto moments is exist is k-2 < 0 = k < 2

That is only we find the first moment for k

… (94 more words) …