List of top Statistics Questions asked in Common University Entrance Test

A random variable X has a distribution with density function
\( f(x;\beta) = \begin{cases} (\beta+1)x^\beta; & 0<x<1; \beta>-1 \\ 0; & \text{otherwise} \end{cases} \) Based on 'n' observations on X, Maximum Likelihood Estimator (MLE) of \(\beta\) is

Consider a 2x2 matrix \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\). If \(a+d=1\) and \(ad-bc=1\), then \(A^3\) is equal to

The system of equations given by \( \begin{bmatrix} 1 & 1 & 1 & : & 3 \\ 0 & -2 & -2 & : & 4 \\ 1 & -5 & 0 & : & 5 \end{bmatrix} \) has the solution:

Consider the probability density function \( f(x;\theta) = \begin{cases} \frac{2x}{5\theta} & ; 0 \le x \le \theta \\ \frac{2(5-x)}{5(5-\theta)} & ; \theta \le x \le 5 \end{cases} \) For a sample of size 3, let the observations are, \( x_1 = 1, x_2 = 4, x_3 = 2 \). Then, the value of likelihood function at \( \theta=2 \) is

If, \(1 \le x \le 1.5\) is the critical region for testing the null hypothesis \(H_0: \theta=1\) against the alternative hypothesis \(H_1: \theta=2\) on the basis of a single observation from the population, \( f(x;\theta) = \begin{cases} \frac{1}{\theta} & ; 0 \le x \le \theta \\ 0 & ; \text{otherwise} \end{cases} \), then the power of the test, is

Let X have a probability density function of the form, \( f(x;\theta) = \begin{cases} \frac{1}{\theta} e^{-x/\theta} & ; 0<x<\infty, \theta>0 \\ 0 & ; \text{otherwise} \end{cases} \) To test null hypothesis \(H_0: \theta = 2\) against the alternate hypothesis \(H_1: \theta = 1\), a random sample of size 2 is taken. For the critical region \(W_0 = \{(x_1, x_2) : 6.5 \le x_1 + x_2\}\), the power of the test is

If X is a random variable such that,
\(P(X \le x) = \begin{cases} 0 & ; x<0 \\ 1-e^{-x\theta} & ; x \ge 0 \end{cases}\)
based on 'n' independent observations on X, the Maximum Likelihood Estimator (MLE) of E(X) is

The regression coefficient of Mumbai prices over Kolkata prices from the following table, is:

	Mumbai (₹)	Kolkata (₹)
Average price (per 5 kg)	120	130
S.D.	4	5
Correlation coefficient	0.6
N (Sample size)	100

From a set of data involving four "tropical feed stuffs A, B, C and D", tried on 20 chicks, the following information was extracted:

\[ \begin{array}{|l|c|c|} \hline \textbf{Source of variation} & \textbf{Sum of squares} & \textbf{Degrees of freedom} \\ \hline \text{Treatment} & 26000 & 3 \\ \text{Error} & 11500 & 16 \\ \hline \end{array} \]

All the 20 chicks were treated alike, except for the feeding treatment, and each feeding treatment was given to 5 chicks. Then, the critical difference between any two means is:

It is given that there are six treatments and four blocks,

\[ \begin{array}{|l|cccccc|} \hline \textbf{Treatment totals} & T_1 & T_2 & T_3 & T_4 & T_5 & T_6 \\ & 63 & 65 & 57 & 64 & 65 & 66 \\ \hline \textbf{Block totals} & B_1 & B_2 & B_3 & B_4 & & \\ & 90 & 85 & 106 & 98 & & \\ \hline \end{array} \]

and that \( G = \sum_i \sum_j y_{ij} = 380 \), then the sum of squares due to treatment is:

For the given ANOVA table:

\[ \begin{array}{|l|c|c|} \hline \textbf{Source of variation} & \textbf{Sum of squares} & \textbf{Degrees of freedom} \\ \hline \text{Service station} & 6810 & 9 \\ \text{Rating} & 400 & 4 \\ \text{Total} & 9948 & 49 \\ \hline \end{array} \]

The test statistic to test that there is no significant difference between the service stations is:

For the given model \(x_{ijk} = \mu + \alpha_i + \beta_j + \gamma_{ij} + e_{ijk}; i=1, \dots,p; j=1, \dots,q; k=1, \dots,m\), the degrees of freedom corresponding to sum of squares due to error is:

In measuring reaction times, a psychologist estimates that the standard deviation is 0.05 seconds. How large a sample of measurements should be taken in order to be 95% confident that the error of the estimate will not exceed 0.01 seconds?

If the standard deviation of marks obtained by 150 students is 11.9, then the standard error of the estimate of the population mean for a random sample of size 30 with SRSWOR, is:

For the given model \(x_{ijk} = \mu + \alpha_i + \beta_j + \gamma_{ij} + e_{ijk}; i=1, \dots,p; j=1, \dots,q; k=1, \dots,m\), under the assumption of the normality of the parent population, the p.d.f. of \(y = \frac{S_{AB}^2}{\sigma_e^2}\), is

Minimum number of replications required, when the coefficient of the variation for the plot values is given to be 12%, for an observed difference of 10% among the sample means to be significant at 5% level, is

If, \(f(x, y) = xe^{-x(y+1)}; x \ge 0, y \ge 0\), then \(E(Y|X = x)\) is

For two random variables X and Y having the joint probability density function \( f(x,y) = \frac{1}{3}(x+y); 0 \le x \le 1, 0 \le y \le 2 \), then cov(X, Y) is

If \(n_i \propto N_i\) and \(p_i = \frac{N_i}{N}\) and k is the number of strata and \(N_i\) is the number of units in the \(i^{th}\) stratum then, Var(\(\bar{y}_{stratified}\)) is:

From the data relating to the yield of dry bark (\(x_1\)), height (\(x_2\)) and girth (\(x_3\)) for 18 cinchona plants, the correlation coefficient are obtained as \(r_{12}=0.77, r_{13} = 0.72, r_{23} = 0.52\). Then, the multiple correlation coefficient \(R_{1.23}\) is

If under SRSWOR, \(U = \sum_{i=1}^{n_1} y_i = n_1 \bar{y}_1\) and \(V = \sum_{j=n_1+1}^{n} y_j = (n-n_1)\bar{y}_2\), then the Var(V) is

If all the zero order correlation coefficients in a set of n-variates are equal to \(\rho\), then every third order partial correlation coefficient is equal to:

If the two regression lines are given by \(8X-10Y+66=0\) and \(40X-18Y=264\), then the correlation coefficient between X and Y is: