ISS Statistics Paper I (Old Subjective Pattern): Questions 1 - 7 of 165

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Question number: 1

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Show that the square of the one sample t-statistic has the F-distribution. What are its degrees of freedom?

Explanation

The t-statistic is defined as the ratio of a standard normal variable X~N (0, 1) and the square root of Y/n where Y~ χn2 and n is the degree of freedom.

t=XYn

Then we show the square of t-statistic follows F-distribution.

t… (430 more words) …

Question number: 2

» Statistical Methods » Regression » Linear

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Show that the best predictor of Y, in terms of minimum MSE, is linear in X, if (X, Y) has bivariate normal distribution.

Explanation

We have two random variables X and Y. We use the value of X to predict Y and (X, Y) has bivariate normal distribution. The correlation coefficient is ρ=Corr (X, Y). Let

(X,Y)~BN(μX,μY,σX2… (416 more words) …

Question number: 3

» Statistical Methods » Tests of Significance » Chi-Square

Appeared in Year: 2013

Essay Question▾

Describe in Detail

From along series of annual river flows, the variance is found out to be 49 units. For a new sample of 25 years, the variance is calculated as 81 units. Can we regard that the sample variance is significant? (Given the chi-square value at 5 % level of significance as 37.7)

Explanation

The sample size is 25. Here we test

H0:σ2=σ02vsHA:σ2σ02

The null hypothesis is tested by chi-square test when the sample size is less than 30. The test statistic is

χ2… (113 more words) …

Question number: 4

» Probability » Central Limit Theorems

Appeared in Year: 2013

Essay Question▾

Describe in Detail

State and prove Lindeberg-Levy Central limit theorem.

Explanation

Lindeberg-Levy Central limit theorem.

Let Y 1, Y 2, …, Y n be independent and identically distributed random variables with common mean

E (Y i) =µ and finite positive variance Var (Y i) = σ 2 for i = 1, 2, …, n

then n… (172 more words) …

Question number: 5

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2013

Essay Question▾

Describe in Detail

If X ~ N (0, 1), obtain the distribution of X 2.

Explanation

X ~ N (0, 1). The density function is

f(x)=12πex22

Let assume Y = X 2

F(y)=P[Yy]=P[X2y]

=P[… (139 more words) …

Question number: 6

» Statistical Methods » Measures of Location

Appeared in Year: 2010

Essay Question▾

Describe in Detail

A cyclist pedals from his house to his college at a speed of l0 km per hour and back from the college to his house at 15 km per hour. Find the average speed.

Explanation

Given that a cyclist pedals from his house to his college at a speed of l0 km per hour and back from the college to his house at 15 km per hour. Assume that the distance between the house to college is d. Here the average speed is finding by… (90 more words) …

Question number: 7

» Statistical Methods » Non-Parametric Test » Wald-Wolfowitz

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Explain the Wald-Wolfowitz run test for randomness in a sequence of two types of symbols. Find E Ho (R) where R denotes the number of runs of elements of one kind.

Explanation

Suppose we have two sample x 1, x 2, …x n and y 1, y 2, …, y m and we wish to test that either both sample come from same population or not. We can use Wald-Wolfowitz run test for randomness.

First we arrange n… (242 more words) …

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