List of top Testing of Hypotheses Questions asked in GATE ST

Let $X_1,X_2,X_3,X_4$ be a sample of size 4 from Bernoulli($\theta$), $0<\theta<1$. We test $H_0:\theta=\tfrac14$ vs $H_1:\theta>\tfrac14$, rejecting $H_0$ if $X_1+X_2+X_3+X_4>2$. If $\alpha$ is the Type-I error and $\gamma(\theta)$ the power function, then find $16\alpha+7\gamma\!\left(\tfrac12\right)$ (in integer).

Given $\Phi(1.645)=0.95$ and $\Phi(2.33)=0.99$. For a random sample $X_1,\ldots,X_n\sim N(\mu,2^2)$ with $\mu$ unknown, test $H_0:\mu=10$ vs $H_1:\mu=12$ using a rule that rejects $H_0$ when $\overline X$ is large. Find the smallest $n$ (in integer) so that Type I error is $0.05$ and Type II error is at most $0.01$.

A random sample $ X $ of size one is taken from a distribution with the probability density function \[ f(x; \theta) = \begin{cases} \frac{2x}{\theta^2}, & 0<x<\theta, \\ 0, & \text{elsewhere.} \end{cases} \] If $ \frac{X}{\theta} $ is used as a pivot for obtaining the confidence interval for $ \theta $, then which one of the following is an 80% confidence interval (confidence limits rounded off to three decimal places) for $ \theta $ based on the observed sample value $ x = 10 $?

Let $ X_1, X_2, \dots, X_7 $ be a random sample from a normal population with mean 0 and variance $ \theta>0 $. Let \[ K = \frac{X_1^2 + X_2^2 + \dots + X_7^2}{X_1^2 + X_2^2 + \dots + X_7^2}. \] Consider the following statements: \begin{enumerate} \item The statistics $ K $ and $ X_1^2 + X_2^2 + \dots + X_7^2 $ are independent. \item $ \frac{7K}{2} $ has an $ F $-distribution with 2 and 7 degrees of freedom. \item $ E(K^2) = \frac{8}{63} $. \end{enumerate} Then which of the above statements is/are true?

A random sample $ X_1, X_2, \dots, X_6 $ is taken from a Bernoulli distribution with the parameter $ \theta $. The null hypothesis $ H_0: \theta = \frac{1}{2} $ is to be tested against the alternative hypothesis $ H_1: \theta>\frac{1}{2} $, based on the statistic $ Y = \sum_{i=1}^{6} X_i $. If the value of $ Y $ corresponding to the observed sample values is 4, then the p-value of the test is ________?