List of top Statistics Questions

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:

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:

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

If \(f(x)\) and \(g(x)\) are differentiable functions for \(0 \leq x \leq 1\) such that, \(f(1)-f(0) = k(g(1)-g(0))\), \(k \neq 0\), and there exists a 'c' satisfying \(0<c<1\). Then, the value of \(\frac{f'(c)}{g'(c)}\) is equal to

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 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:

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 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, \(X \sim N(\theta, 1)\) 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 \(Z_\alpha = 1.64\))

Let, random variable \(X \sim \text{Bernoulli}(p)\). Then, \(\beta_1\) 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

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:

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?

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

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

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

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

It is given that at x = 1, the function \(f(x) = x^4 - 62x^2 + ax + 9\), attains its maximum value in the interval \([0, 2]\). Then, the value of 'a' is

If, \(y = x^{\tan(x)}\), then \(\frac{dy}{dx}\) at \(x = \frac{\pi}{4}\), is