# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 1 - 7 of 164

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## Question 1

Standard Probability Distributions
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Appeared in Year: 2011

### Describe in Detail

Essay▾

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 where Y ~ and n is the degree of freedom.

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

We known that X is standard normal distribution, then X 2 follows a chi-square distribution with degree of freedom is 1. Let assume Z =

The jo&#8230;

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## Question 2

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Appeared in Year: 2011

### Describe in Detail

Essay▾

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

We first suppose the linear function of prediction is a + bX. Then the mean square error is given by

Then the point for which the MSE of Y is minimum by solving simultaneous equati&#8230;

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## Question 3

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Appeared in Year: 2013

### Describe in Detail

Essay▾

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

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

The decision criteria is reject the null hypothesis if the calculated value is greater than the tabulated value otherwise accept it.

Here n = 25, ,

The tabulated value at 5 % level of significance is 37.7

So,&#8230;

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## Question 4

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Appeared in Year: 2013

### Describe in Detail

Essay▾

State and prove Lindeberg-Levy Central limit theorem.

### Explanation

Lindeberg-Levy Central limit theorem.

Let Y1 , Y2 , … , Yn be independent and identically distributed random variables with common mean

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

then , the distribution of the um of these random variables

Sn = Y1 + Y2 + … + Yn

tends to the normal distribution with mean n µ and variance nσ&#8230;

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## Question 5

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Appeared in Year: 2013

### Describe in Detail

Essay▾

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

### Explanation

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

Let assume Y = X 2

Let

The limit is also change

Differentiate with respect to y,

So, Y = X 2 is follows a chi-square distribution with one degree of freedom.

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## Question 6

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Appeared in Year: 2010

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Essay▾

A cyclist pedals from his house to his college at a speed of 10 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 10 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 harmonic mean because it is suitable when the values are pertaining to the rate of change per unit time such&#8230;

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## Question 7

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Appeared in Year: 2011

### Describe in Detail

Essay▾

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

### Explanation

Suppose we have two sample x1 , x2 , … xn and y1 , y2 , … , ym 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 x՚s and m y՚s in descending order of size (n + m) . Then we consider the array of x՚s and y՚s and count the number of runs (R) . A run is a &#8230;

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