List of top Statistics Questions

A box contains 8 balls, \(\theta\) of which are white. The null hypothesis \(H_0: \theta = 3\) is tested by drawing 2 balls at random without replacement.

The hypothesis is rejected in favour of the alternative hypothesis \(H_1: \theta > 3\) if both balls are white.

What is the significance level of the test?

Match List - I with List - II.

List - I	List - II
A. Minimum variance bound estimator	I. Cramer-Rao Inequality
B. Sufficient statistics	II. Factorisation Theorem
C. UMVU estimator	III. Rao-Blackwell Theorem
D. Unique UMVUE	IV. Complete sufficient statistics

A test based on critical region \(W\) is said to be unbiased for testing \(H_0: \theta = \theta_0\) against \(H_1: \theta \ne \theta_0\) if (where \(\alpha\) and \(\beta\) are Type I and Type II errors respectively)

If \( A_{x:n} = \{k : |k - \tfrac{n}{2}| \le \tfrac{x\sqrt{n}}{2} \} \), then the value of

\( \lim_{n \to \infty} \sum_{k \in A_{1:n}} \binom{n}{k} 2^{-n} \)

(where \( \Phi(\cdot) \) is the distribution function of the standard normal variate) is:

The random variable \(Y \sim U(0, X)\), where the marginal density of \(X\) isv

\( f_X(x) = \begin{cases} 2x & \text{for } 0 < x < 1 \\ 0 & \text{otherwise} \end{cases} \)

Then \(E(Y)\) is:

A random variable \(X\) has a cumulative distribution function \(F_X(x)\) given by

\( F_X(x) = \begin{cases} 0 & \text{if } x < 1 \\ \frac{x^{2} - 2x + 2}{2} & \text{if } 1 \le x < 2 \\ 1 & \text{if } x \ge 2 \end{cases} \)

Then \(E(X)\) is:

The area of bounded region R defined as \( R = \{(x,y): 0 < x < 2,\; 1 < y < 3,\; y > x \} \) is

A simple random sample of size 'n' will be drawn from a class of 125 students, and mean mathematics score of the sample will be compute
D. If the standard error of the sample mean for "with replacement sampling" is twice as much as the standard error of the sample mean for "without replacement sampling", then the value of n is

To test whether two different skin creams A and B have different effects on the human body, $n$ persons were enrolled in a clinical trial. Then cream A was applied to one arm and cream B to the other arm of each of the $n$ randomly selected persons. What type of design is this?

In a randomized block design with one factor having 5 levels and another factor having 5 levels, the degree of freedom for the error sum of squares are equal to

The method of moment estimator of parameter '$\alpha$' in the following probability density function $f_{X}(x)=\begin{cases}\frac{\alpha}{x^{\alpha+1}}, & x>1, \alpha>1 \\0 & \text{otherwise}\end{cases}$ is

If $X_{1}, X_{2}, \dots, X_{n}$ be independent random variable each from Gamma($\alpha, \beta$) then the jointly sufficient statistics for vector ($\alpha, \beta$) is

Suppose that $X_{1}, X_{2}, \dots, X_{n}$ are independent random variables each drawn from a population having density function $f_{X}(x)=\begin{cases} \frac{1}{\theta} e^{-(x-\mu)/\theta} & \text{if } x \ge \mu \\ 0 & x < \mu \end{cases}$ where $\theta > 0$ and $\mu \in R^{+}$, then maximum likelihood estimate of $(\theta, \mu)$, when both $\theta, \mu$ are unknown is

If $X_{1}=17, X_{2}=10, X_{3}=32$ and $X_{4}=5$ be the observed values of a random sample from the discrete distribution $P(X=x)=\begin{cases}\frac{\theta^{2x} e^{-\theta^2}}{x!} & \text{if } x=0,1,2,\dots, \theta>0 \\ 0 & \text{otherwise}\end{cases}$. Then MLE is

Let $X_{1}, X_{2}, \dots, X_{n}$ constitute a random sample of size "n" from a population having density $f_{X}(x)=\begin{cases} e^{-(x-\theta)} & x > \theta \\ 0 & \text{otherwise} \end{cases}$ then
A. $X_{(1)}$ is sufficient for $\theta$

B. $X_{(1)}$ is consistent for $\theta$

C. $X_{(1)}$ is unbiased for $\theta$

D. $MSE(X_{(1)}) = \frac{2}{n^2}$

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from $U(\theta, \theta+1)$ then the maximum likelihood estimate of $\theta$

Let $X_{1}, X_{2}, \dots, X_{n}$ be random sample from a Poisson family with parameter $\lambda$. Then the maximum likelihood estimate of $P(X \ge 2)$ is

Value of $\lim_{n \rightarrow \infty} e^{-n}\left(1+n+\frac{n^{2}}{2 !}+\dots+\frac{n^{n}}{n !}\right)$ is

Let $X_{i} \sim N(0,1), i=1,2, \dots$ be independent random variables. If $T_{n}=\sum_{i=1}^{n} X_{i}^{2}$, then $\lim_{n \rightarrow \infty} P(T_{n} > n+2 \sqrt{2 n})$ is equal to

If $X_{1}, X_{2}, \dots, X_{n}$ denote a random sample of size $n$ from normal population $N(0, \theta^{2})$ then MVUE of $\theta^{2}$ is

Let $\{X_{n}\}$ be a sequence of $r \cdot v$'s and $Y_{n}=\left(\frac{S_{n}-E(S_{n})}{n}\right)$ where $S_{n}=\sum_{i=1}^{n} X_{i}$ then the necessary and sufficient condition for the sequence $\{X_{n}\}$ to satisfy W.L.L.N is

The joint density function of X and Y is $f_{XY}(x,y)=\begin{cases}x+y & \text{for } 0<x<1, 0<y<1 \\ 0 & \text{elsewhere}\end{cases}$ then $P(X<2Y)$ is

Consider the regression model $y_{i}=\beta_{0}+i\beta_{1}+\epsilon_{i} (i=1,2,...,n>2)$ where $\beta_{0}$ and $\beta_{1}$ are unknown parameters and $\epsilon_{i}$ are random errors. Let $y_{i}$ be the observed value of $Y_{i}(i=1,2,...,n)$. Using the method of ordinary least squares, the estimate of $\beta_{1}$ is

Let $X_{1}, X_{2}, \dots$ be independent variables each taking values $+1$ or $-1$ with equal probability respectively. If $S_{n}=\sum_{i=1}^{n} i X_{i}$ then $\lim_{n \rightarrow \infty} P\left(S_{n} < \sqrt{\frac{n(n+1)(2 n+1)}{3}}\right)$ where $\Phi$ is distribution function of standard normal variate, is

Let $X$ be a random variable having probability density function $f_{X}(x)=\begin{cases}2x & 0<x<1 \\ 0 & \text{otherwise}\end{cases}$ then the density of $Y=\frac{1}{X^{\frac{1}{\alpha}}}$ is

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