List of practice Questions

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 \(X_1, X_2, X_3, X_4\) 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

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

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

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

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

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

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

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

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 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 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 \(X_1, X_2, \dots, X_n\) 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

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:

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 man buys 60 electric bulbs from a company "P" and 70 bulbs from another company, "H". He finds that the average life of P's bulbs is 1500 hours with a standard deviation of 60 hours and the average life of H's bulbs is 1550 hours with a standard deviation of 70 hours. Then, the value of the test statistic to test that there is no significant difference between the mean lives of bulbs from the two companies, 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 \(t_{0.05} (25) = 2.06\))

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

Let \(X_1, X_2, X_3\) be three variables with means 3, 4 and 5 respectively, variances 10, 20 and 30 respectively and \(cov (X_1, X_2) = cov (X_2, X_3) = 0\) and \(cov (X_1, X_3) = 5\). If, \(Y = 2X_1 +3X_2+4X_3\) then, Var(\(Y\)) 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