List of practice Questions

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:

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:

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?

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

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

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

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 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 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 \(n_i \propto N_i\) 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:

If \(X \sim \beta_1(\alpha, \beta)\) such that parameters \(\alpha, \beta\) are unknown, then the sufficient statistic for \((\alpha, \beta)\) 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 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, \(U = \frac{X-a}{h}\), \(V = \frac{Y-b}{k}\); \(a, b, h, k>0\), then \(b_{UV}\) is

The correlation coefficient between two variables X and Y is 0.60 and it is given that \(\sigma_X = 2, \sigma_Y = 4\). Then, the angle between two lines of regression, is

If \(f(X) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}; -\infty<x<\infty\) and \(Y = |X|\), then E(Y) 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 \(X_1, X_2, X_3, X_4\) 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 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:

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

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

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