# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 41 - 46 of 165

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## Question number: 41

» Numerical Analysis » Interpolation Formulae » Lagrange

Appeared in Year: 2013

### Describe in Detail

Find the missing value in the table by assuming a polynomial form for y:

X | 0 | 1 | 2 | 3 | 4 |

y | 1 | 2 | 4 | - | 16 |

### Explanation

we have given that there are four values of y is given, then find the missing value of y at x = 3 using Lagrange’s interpolation polynomial because the x values is not equal interval.

The table is

X | 0 | 1 | 2 | 4 |

y | 1 | 2 | 4 | 16 |

## Question number: 42

» Probability » Tchebycheffs Inequality

Appeared in Year: 2013

### Describe in Detail

Let X be a random variable with E [X] = 4 and E [X ^{2}] = 20. Use Chebyshev’s inequality to determine a lower bound for the probability P [0 < x < 8].

### Explanation

Let X be a random variable with mean µ and variance σ ^{2}. Then any k > 0, the Chebyshev’s inequality is

or

σ ^{2} = E [X ^{2}] - (E [X] ) ^{2} =4

Then, a lower bound for the probability

Using Chebyshev’s inequality

## Question number: 43

» Probability » Probability Generating Functions

Appeared in Year: 2014

### Describe in Detail

Let P (s) be the probability generating function associated with a non-negative, integer valued random variable X. Show that

### Explanation

To show this we use the left hand side, express the summation term

We know that the probability generating function is defined as

Putting this in the above equation, we prove the following

## Question number: 44

» Probability » Bayes' Theorem

Appeared in Year: 2014

### Describe in Detail

There are three identical bags U _{1}, U _{2} and U _{3}. U _{1} contains 3 red and 4 black balls; U _{2} contains 4 red and 5 black balls; U _{3} contains 4 red and 4 black balls. One bag is chosen at random; a ball is drawn at random from the chosen bag and it is found to be red. Find the probability that the first bag is chosen.

### Explanation

Given that there are three identical bags U _{1}, U _{2} and U _{3}. Then probabilities of selecting a bag are,

Let X be the event of selecting a red ball

Probability of selecting a red ball in U _{1} is

Similarly, Probability of selecting a

## Question number: 45

» Statistical Methods » Measures of Location

Appeared in Year: 2010

### Describe in Detail

Show that for any discrete distribution, standard deviation is not less than mean deviation from mean.

### Explanation

Here, we show that standard deviation is greater than mean deviation from mean.

S. D. ≥ mean deviation from mean

(S. D. ) ^{2} ≥ (mean deviation from mean) ^{2}

Let x _{1}, x _{2}, …, x _{n} are n observations in a set have frequency f _{1}

## Question number: 46

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

Appeared in Year: 2014

### Describe in Detail

Consider two samples as follows:

Sample 1: 1, 4, 7, 9, 16, 17, 22, 24

Sample 2: 2, 6, 10, 12, 18, 20, 26, 28, 32

Test whether the examples have come from the same population by Wilcoxon-Mann-Whitney test. [Given value of Z for α = 0.05 = 1.645, where Z is N (0, 1) ]

### Explanation

Let the sample 1 consider the population X and sample 2 consider the population Y. The hypothesis whether two sample comes from same identical population. We test the following null and alternative hypothesis is

In Wilcoxon-Mann-Whitney test, first consider the combined ordered sequence of the sample values is

Sample: