List of top Statistics Questions asked in Common University Entrance Test

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:

If, \(X \sim N(\theta, 1)\) and in order to test \(H_0: \theta=1\) against the alternate \(H_1: \theta=2\) a random sample \((x_1, x_2)\) of size 2 is taken. Then, the best critical region (B.C.R.) is given by (where \(Z_\alpha = 1.64\))

If \(f(X) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}; -\infty<x<\infty\) and \(Y = |X|\), then E(Y) is

Let \(X_1, X_2, X_3, X_4\) be a sample of size 4 from a U(0,\(\theta\)) distribution. Suppose that, in order to test the hypothesis \(H_0: \theta = 1\) against the alternate \(H_1: \theta \ne 1\), an UMPCR is given by, \(W_0 = \{x_{(4)} : x_{(4)}<\frac{1}{2} \text{ or } x_{(4)}>1\}\), then the size \(\alpha\) of \(W_0\) is

Let \( \hat{\lambda} \) be the Maximum Likelihood Estimator of the parameter \(\lambda\), then, on the basis of a sample of size 'n' from a population having the probability density function \( f(x; \lambda) = \frac{e^{-\lambda} \lambda^x}{x!} \); \(x = 0, 1, 2, \dots\), \(\lambda>0\), the Var(\(\hat{\lambda}\)) is

Let p be the probability that a coin will fall heads in a single toss in order to test \(H_0: p = \frac{1}{2}\) against the alternate \(H_1: p = \frac{3}{4}\). The coin is tossed five times and \(H_0\) is rejected if 3 or more than 3 heads are obtained. Then, the probability of Type I error, is

If, \(x \ge 1\) is the critical region for testing \(H_0: \theta = 2\) against the alternate \(H_1: \theta = 1\). On the basis of a single observation from the population \(f(x;\theta) = \theta e^{-x\theta}; x>0, \theta>0\), then the size of Type II error is:

In a simple random sample of 600 people taken from a city A, 400 smoke. In another sample of 900 people taken from a city B, 450 smoke. Then, the value of the test statistic to test the difference between the proportions of smokers in the two samples, is:

Let, X and Y be independent and identically distributed Poisson(1) variables. If, Z = min(X, Y) then, P(Z = 1) is:

If, \(X \sim \text{Bin}(8, 1/2)\) and \(Y = X^2+2\), then \(P(Y \le 6)\) is:

Moment generating function of a random variable Y, is \( \frac{1}{3}e^t(e^t - \frac{2}{3}) \), then E(Y) is given by

If, \(f(X) = \frac{C\theta^x}{x}\); \(x = 1,2, \dots\); \(0<\theta<1\), then E(X) is equal to

If, \(f(x; \alpha, \beta) = \begin{cases} \alpha \beta x^{\beta-1} e^{-\alpha x^\beta} & ; x>0 \text{ and } \alpha, \beta>0 \\ 0 & ; \text{otherwise} \end{cases}\), then the probability density function of \(Y=x^\beta\) is

In a binomial distribution consisting of five independent trails, the probability of 1 and 2 success are 0.4096 and 0.2048 respectively. Then, the parameter 'p' of distribution is

If X and Y are independent and identically distributed geometric variables with parameter p, then the moment generating function of (X+Y) is given by

Let, \(X \sim \beta_1(u, v)\) and \(Y \sim \gamma(1, u+v)\); (\(u, v>0\)) be independent random variables. If, \(Z = XY\), then the moment generating function of Z is given by

In a hypothetical group, it is given that \( d = 0.05 \), \( p=0.5\alpha \) and \( t = 2 \). If N is large, then the sample size \( n_0 \), is

Mean height of plants obtained from a random sample of size 100 is 64 inches. The population standard deviation of the plants is 3 inches. If the plant heights are distributed normally, then the 99% confidence limits of the mean population height of plants, are:

A sample of size 1600 is taken from a population of fathers and sons and the correlation between their heights is found to be 0.80. Then, the correlation limits for the entire population are:

If \(X_1, X_2, \dots, X_n\) is a random sample from the population \(f(x, \theta) = (\theta+1)x^\theta; 0<x<1; \theta>-1\) and \(Y = -\sum_{i=1}^{n} \log(x_i)\). Then \(E\left(\frac{1}{Y}\right)\) is

In a survey of 200 boys, 75 were intelligent and out of these intelligent boys, 40 had an education from the government schools. Out of not intelligent boys, 85 had an education form the private schools. Then, the value of the test statistic, to test the hypothesis that there is no association between the education from the schools and intelligence of boys, is:

If, joint distribution function of two random variables X and Y is given by \(F_{X,Y}(x,y) = \begin{cases} 1 - e^{-x} - e^{-y} + e^{-(x+y)} & ; x>0; y>0 \\ 0 & ; \text{otherwise} \end{cases}\), then Var(\(X\)) is

Minimum value of the correlation coefficient 'r' in a sample of 27 pairs from a bivariate normal population, significant at 5% level, is: (Given \(t_{0.05} (25) = 2.06\))

Out of 800 families with 4 children each, the percentage of families having no girls is:

Three urns contain 3 green and 2 white balls, 5 green and 6 white balls and 2 green and 4 white balls respectively. One ball is drawn at random from each of the urn. Then, the expected number of white balls drawn, is