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Model Test Paper, Mathematical Statistics - 2017 - Question 1

If (x_{n}) is a sequence of real numbers which converges to x then the sequence (s_{n}) where

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Model Test Paper, Mathematical Statistics - 2017 - Question 2

The general solution of the differential equation y”(x) – 4y’(x) + 8y(x) + 10e^{x }cos x is

Model Test Paper, Mathematical Statistics - 2017 - Question 3

If the probability that A and B will die within a year are p and q respectively, then the probability that only one of them will be alive at the end of the year is

Model Test Paper, Mathematical Statistics - 2017 - Question 4

The differential equation 2ydx – (3y – 2x)dy = 0 is

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Model Test Paper, Mathematical Statistics - 2017 - Question 5

What should be the value of z used in a 93% confidence interval?

Model Test Paper, Mathematical Statistics - 2017 - Question 6

Let X be a discrete random variable with values x = 0, 1, 2 and probabilities P(X = 0) = 0.25, P(X = 1) = 0.50, and P(X = 2) = 0.25, respectively.Find E(X^{2)}

Model Test Paper, Mathematical Statistics - 2017 - Question 7

If two dice are thrown, what is the expected value of sum of the face values?

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Model Test Paper, Mathematical Statistics - 2017 - Question 9

Let a_{n} be a sequence such that a_{1} = a, a_{2} = b and an = (a + a_{n–1})/2 for n > 2. Calculate the limit?

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Model Test Paper, Mathematical Statistics - 2017 - Question 11

A function f is defined on interval (0, 1) as follows

then which one of the following is true?

Model Test Paper, Mathematical Statistics - 2017 - Question 12

Let f : R → R be s.t. f(x – f(y) ≤ (x – y)^{2} then which of the following is TRUE ?

Model Test Paper, Mathematical Statistics - 2017 - Question 13

If two events A and B are such that P(A^{c}) = 0.3, P(B) = 0.4, P(A ∩ B^{c}) = 0.5, then P(B/A ∪ B^{c}) =

Model Test Paper, Mathematical Statistics - 2017 - Question 14

A die is thrown (n + 2) times. After each throw a ‘+’ is recorded for 4, 5 or 6 and ‘–’ for 1, 2 or 3, the signs forming an ordered sequence each, except the first and the last sign, is attached a characteristic random variable which takes the value 1 if both the neighbouring signs differ from the one between them and 0 otherwise. If X_{1}, X_{2}, ..., X_{n} are characteristic random variables, find the mean and variance of

Model Test Paper, Mathematical Statistics - 2017 - Question 15

In the regression line Y = a + bX:

Model Test Paper, Mathematical Statistics - 2017 - Question 16

What would a chi- square significance value of P > 0.05 suggest?

Model Test Paper, Mathematical Statistics - 2017 - Question 17

A random variable X has the density function f(x) = c/(x^{2} + 1). where –∞ < x < ∞. Find the probability that X^{2} lies between 1/3 and 1.

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Model Test Paper, Mathematical Statistics - 2017 - Question 18

Y is an exponential random variable with parameter λ = 0.2. Given the event A = {Y < 2}.

Find the conditional expected value, E [Y | A].

Model Test Paper, Mathematical Statistics - 2017 - Question 19

An examination paper has 150 multiple- choice questions of one mark each, with each question having four choice. Each incorrect answer fetches- 0.25 mark. Suppose 1000 students choose all their answers randomly with uniform probability. The sum total of the expected marks obtained by all these students is :

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Model Test Paper, Mathematical Statistics - 2017 - Question 20

If probability density function of a random variable X is

f(x) = x^{2 }for –1 < x < 1, and

= 0 for any other value of x

then, the percentage probability is

Model Test Paper, Mathematical Statistics - 2017 - Question 21

Let F(x, y) be the d.f. of X and Y

if : F(x, y) = 1, for x + 2y ≥ 1

F(x, y) = 0, for x + 2y < 1,

then

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Model Test Paper, Mathematical Statistics - 2017 - Question 22

If the product moment of X and Y is 3 and the mean of X and Y are both equal to 2, then what is the covariance of the random variables 2X + 10 and – 5/2Y + 3?

Model Test Paper, Mathematical Statistics - 2017 - Question 23

If X ~ N(μ, σ^{2}) and X_{1}, X_{2}, ..., X_{n} be a random sample from the population X, then

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Model Test Paper, Mathematical Statistics - 2017 - Question 24

Which of the following are true statements?

I. The are under the curve of the t- distribution between ± 1 standard deviation is greater when d.f. = 5 than when d.f. = 10.

II. There is less are in the tails, beyond ± 3 standard deviations, of t-distribution when d.f. = 5 than when d.f. = 10.

III. For a given α, the critical t- value increases as d.f. decreases.

Model Test Paper, Mathematical Statistics - 2017 - Question 25

American Airlines claims that the average number of people who pay for in- flight moves, when the plane is fully loaded, is 42 with a standard deviation of 8. A sample of 36 fully loaded planes is taken. What is the probability that fewer than 38 people paid for the in- flight moves?

Model Test Paper, Mathematical Statistics - 2017 - Question 26

A symmetric die is thrown 600 times. Find the lower hound for the probability of getting 80 to 120 sixes.

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Model Test Paper, Mathematical Statistics - 2017 - Question 27

An estimator T_{1} = t_{1}(x_{1}, x_{2}, ..., x_{n}) for q is said to be admissible if for any other estimator T_{2} = t_{2}(x_{1}, x_{2}, ..., x_{n}) for q, the relation is of the type:

Model Test Paper, Mathematical Statistics - 2017 - Question 28

A sample of 3 observations, (X_{1} = 0.4, X_{2} = 0.7, X_{3} = 0.9) is collected from a continuous distribution with density

Estimate θ by the method of moments;

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Model Test Paper, Mathematical Statistics - 2017 - Question 29

In a test of H_{0} : μ = 100 against H_{A} : μ ≠ 100, a sample of size 10 produces a sample mean of 103 and a p- value of 0.08. Thus, at the 0.05 level of significance:

Model Test Paper, Mathematical Statistics - 2017 - Question 30

Suppose n = 100. Then the probability of type II error is :

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Model Test Paper, Mathematical Statistics - 2017 - Question 31

A monotonically increasing sequence <x_{n}> is convergent if

*Multiple options can be correct

Model Test Paper, Mathematical Statistics - 2017 - Question 32

Let Y be a binomial random variable with parameters n = 100 and p = 1/2 Using the Central LimitTheorem which of the following are correct?

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*Multiple options can be correct

Model Test Paper, Mathematical Statistics - 2017 - Question 33

The distribution has two parameters. Given X_{1},....,X_{n} is a random sample from the N(μ, σ^{2}) distribution, find the method of moments estimates of μ and σ^{2}.

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Model Test Paper, Mathematical Statistics - 2017 - Question 34

If f(x) = x sin x for all x ∈ R then f : R → R is

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Model Test Paper, Mathematical Statistics - 2017 - Question 35

Let {X_{n}} be a sequence of independent Bernoulli random variables with parameter p = 1/2

generic term X_{n} of the sequence has support R_{n} = { 0, 1} and probability mass function:

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 35

*Multiple options can be correct

Model Test Paper, Mathematical Statistics - 2017 - Question 36

Cauchy’s nth Root Test

Let ∑u_{n} be a positive term series such that

*Multiple options can be correct

Model Test Paper, Mathematical Statistics - 2017 - Question 37

Which of the following statements holds in (0, 2) if the function y = In (3x^{4} – 2x^{3} – 6x^{2} + 6x + 1)

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Model Test Paper, Mathematical Statistics - 2017 - Question 38

The probability density function of X, the lifetime of a certain type of electronic device (measured in hours), is given by,

then which of the following holds ?

*Multiple options can be correct

Model Test Paper, Mathematical Statistics - 2017 - Question 39

In 360 tosses of a pair of dice, 74 “sevens” and 24 “elevens” are observed then which of the following are correct?

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Model Test Paper, Mathematical Statistics - 2017 - Question 40

Given that lim a_{n} = a, lim b_{n} = b, and <Sn> and <Tn> are two sequences, where S_{n} = max {a_{n}, b_{n}} and T_{n} = min {a_{n}, b_{n}}. then which of the following are true?

*Answer can only contain numeric values

*Answer can only contain numeric values

Model Test Paper, Mathematical Statistics - 2017 - Question 42

The following nonlinear differential equation can be solved exactly by separation of variables.

The value of θ(100) most nearly is

*Answer can only contain numeric values

Model Test Paper, Mathematical Statistics - 2017 - Question 43

Using the Gauss- Jorden reduction method, if we find the inverse of the matrix a_{11 }of the inverted matrix is _______

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Model Test Paper, Mathematical Statistics - 2017 - Question 44

In the integral

change the order of integration, and evaluate the integral.

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Model Test Paper, Mathematical Statistics - 2017 - Question 45

If X follows a binomial distribution with parameters n = 100 p = 1/3, then P(X = r) is maximum when r =

*Answer can only contain numeric values

Model Test Paper, Mathematical Statistics - 2017 - Question 46

The solution for what should be the value of a?

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Model Test Paper, Mathematical Statistics - 2017 - Question 47

Let f:R → R be continuous such that f(x) = x^{2} + 5 for all x ∈ Q then the value of f ( √2 ) is

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Model Test Paper, Mathematical Statistics - 2017 - Question 48

Twenty sheets of aluminum alloy were examined for surface flaws. The frequency of the number of sheets with a given number of flaws per sheet was as follows :

What is the probability of finding a sheet chosen at random which contains 3 or more surface flaws?

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 48

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Model Test Paper, Mathematical Statistics - 2017 - Question 49

You know the population mean for a certain test score. You select 10 people from the population to estimate the standard deviation. How many degrees of freedom does your estimation of the standard deviation have?

*Answer can only contain numeric values

Model Test Paper, Mathematical Statistics - 2017 - Question 50

A waiter believes that his tips from various customers have a slightly right skewed distribution with a mean of 10 dollars and a standard deviation of 2.50 dollars. What is the probability that the average of 35 customers will be more than 13 dollars?

*Answer can only contain numeric values

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 51

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Model Test Paper, Mathematical Statistics - 2017 - Question 52

In psychology research, it is conventional to reject the null hypothesis if the probability value is lower than what number?

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Model Test Paper, Mathematical Statistics - 2017 - Question 53

The distribution only has one parameter. Given X_{1}, X_{2}, ..., X_{n} is a random sample from a U(0, θ) distribution, find the coefficient of method of moments estimator (MMFE) of θ.

*Answer can only contain numeric values

Model Test Paper, Mathematical Statistics - 2017 - Question 54

You do not know the population mean of a different test score. You select 15 people from the population and use this sample to estimate the mean and standard deviation. How many degrees of freedom does your estimation of the standard deviation have?

*Answer can only contain numeric values

Model Test Paper, Mathematical Statistics - 2017 - Question 55

If electricity power failures occur according to a Poisson distribution with an average of 3 failures every twenty weeks,calculate the probability that there will not be more than one failure during a particular week.

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*Answer can only contain numeric values

*Answer can only contain numeric values

Model Test Paper, Mathematical Statistics - 2017 - Question 57

The series converges absolutely to the limit__.

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Model Test Paper, Mathematical Statistics - 2017 - Question 58

From the following table showing the number of plants having certain character, test the hypothesis that the flower colour is independent of flatness of leaves.

You may use the following table giving the value of c2 for one degree of freedom, for different values of P.

Calculate the value of χ^{2} for the above table

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 58

*Answer can only contain numeric values

Model Test Paper, Mathematical Statistics - 2017 - Question 59

Suppose that a netball player has a probability of 1/2 of scoring a goal each time. What is the

probability that she will score one goal from her first two attempts ?

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*Answer can only contain numeric values

Model Test Paper, Mathematical Statistics - 2017 - Question 60

Suppose you observe a sample of 100 independent draws from a normal distribution having known mean μ = 0 and unknown variance σ^{2} Denote the 100 draws by X_{1}, ..., X_{100}. Suppose that:

Find an upper limit of confidence interval for s^{2}, using a set estimator of s^{2} having 99% coverage probability.

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