List of top Statistics Questions on Hypothesis testing

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, \(X \sim N(\theta, 1)\) 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 \(X_1, X_2, X_3, X_4\) 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:

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:

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:

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

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:

Let \( X_1, X_2 \) be a random sample from a population having probability density function
\[ f_{\theta}(x) = \begin{cases} e^{(x-\theta)} & \text{if } -\infty < x \leq \theta, \\ 0 & \text{otherwise}, \end{cases} \] where \( \theta \in \mathbb{R} \) is an unknown parameter. Consider testing \( H_0: \theta \geq 0 \) against \( H_1: \theta < 0 \) at level \( \alpha = 0.09 \). Let \( \beta(\theta) \) denote the power function of a uniformly most powerful test. Then \( \beta(\log_e 0.36) \) equals ________ (rounded off to two decimal places).

Let \( X_1, X_2 \) be a random sample from a distribution having probability density function

Let \( X \) be a random variable having probability density function \( f \in \{ f_0, f_1 \} \). Let \( \phi \) be a most powerful test of level 0.05 for testing \( H_0: f = f_0 \) against \( H_1: f = f_1 \) based on \( X \). Then which one of the following options is NOT necessarily correct?

Let a random variable \( X \) follow a distribution with density \( f \in \{f_0, f_1\} \), where
\[ f_0(x) = \begin{cases} 1 & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise}, \end{cases} \] \[ f_1(x) = \begin{cases} 1 & \text{if } 1 \leq x \leq 2 \\ 0 & \text{otherwise}. \end{cases} \] Let \( \phi \) be a most powerful test of level 0.05 for testing \( H_0: f = f_0 \) against \( H_1: f = f_1 \) based on \( X \). Then which one of the following options is necessarily correct?