# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 25 - 29 of 165

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## Question number: 25

» Statistical Methods » Bivariate Distributions » Bivariate Normal Distribution

Appeared in Year: 2015

### Describe in Detail

Let (X, Y) be distributed as bivariate normal BVN (3, 1, 16, 25, 3/5). Calculate P (4 < Y < 11.84|X=7)

### Explanation

We had known that the conditional distribution of Y given X has mean and variance .

Using conditional mean and variance

## Question number: 26

» Statistical Methods » Tests of Significance » Z-Test

Appeared in Year: 2009

### Describe in Detail

A manufacturer of alkaline batteries expects that only 5 % of his products are defective. A random sample of 300 batteries contained 10 defectives. Can we conclude the proportion of defectives in the entire lot is less than 0.5 at 5 % level of significance?

### Explanation

A random sample of 300 batteries contained 10 defectives. A manufacturer of alkaline batteries expects that only 5 % of his products are defective that is testing of hypothesis is

H _{0}: P = 0.05 against H _{1}: P < 0.05

Here n = 300, numbers of defectives

## Question number: 27

» Statistical Methods » Tests of Significance » T-Test

Appeared in Year: 2010

### Describe in Detail

You are working as a purchase manager for a company. The following information has been supplied to you by two manufacturers of electric bulbs.

Company A | Company B | |

Mean life | 1300 | 1288 |

Standard deviation | 82 | 93 |

Sample size | 100 | 100 |

Which brand of bulbs are you going to purchase if you desire to take a risk of 5%?

### Explanation

Here the sample size is n = 100

The standard deviation and mean of two samples is different. The null hypothesis is the mean life of bulb of company A is equal to the mean life of bulb of company B and alternative its differ.

The test statistic is

## Question number: 28

» Statistical Methods » Measures of Location

Appeared in Year: 2011

### Describe in Detail

Let Z be a random variable with p. d. f. f (z). Let z _{α} be its upper α ^{th} quantile. Show that if X is

a random variable with p. d. f. then σz _{α} +µ is the upper α ^{th} quantile of X.

### Explanation

Let Z be a random variable with p. d. f. f (z). Given that the quantile of z is defined as

X is a random variable with p. d. f. , then quantile is

Assume

Then the limit is

Lower limit is - and upper

## Question number: 29

» Probability » Tchebycheffs Inequality

Appeared in Year: 2014

### Describe in Detail

Show that for 40, 000 throws of a balanced coin, the probability is at least 0.99 that the proportion of heads will fall between 0.475 and 0.525.

### Explanation

For balanced coin the probability is p = 0.5. For Bernoulli trails where n = 40, 000, the mean and standard deviation is

Using Chebyshev’s inequality,

The question says that the probability is at least 0.99 that is

So, the number of heads comes between