# ISS (Statistical Services) 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

### 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 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 freedo

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## Question number: 2

» Statistical Methods » Regression » Linear

Appeared in Year: 2011

### 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

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

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## Question number: 3

» Statistical Methods » Tests of Significance » Chi-Square

Appeared in Year: 2013

### 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

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 =

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## Question number: 4

» Probability » Central Limit Theorems

Appeared in Year: 2013

### 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 , the distribution of the um of these random variables

S _{n} = Y _{1} + Y _{2} +…+Y _{n}

tends to the normal distribution with mean

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## Question number: 5

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2013

### Describe in Detail

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 wit

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## Question number: 6

» Statistical Methods » Measures of Location

Appeared in Year: 2010

### 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 harmonic mean because it is suitable when the values are pertaining to the rate of change per unit time su

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## Question number: 7

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

Appeared in Year: 2011

### 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 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

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