List of top Statistics Questions on Testing of Hypotheses asked in GATE ST

For testing a null hypothesis \(H_0\) against an alternative hypothesis \(H_1\) at level of significance \(\alpha \in (0,1)\), which of the following statements is correct?

Let \(X\) be a single observation from a distribution having the probability density function
\[f_\theta(x)=\begin{cases}1 & \text{if } \theta<x<\theta+1\\0 & \text{otherwise,}\end{cases}\]
where \(\theta\in(-\infty,\infty)\). For testing \(H_0:\theta\le0\) against \(H_1:\theta>1\), let \(\beta\) be the power of the uniformly most powerful test of level \(0.05\). Then \(225\beta\) equals ______ (answer in integer).

Let \(X_1,X_2,\ldots,X_n\ (n\geq2)\) be a random sample from the probability density function \(f(x)\). Consider the following hypotheses:
\[ H_0: f(x)=\frac{1}{\sqrt{2\pi}}\,e^{-\frac{x^2}{2}};\quad -\infty<x<\infty \]
\[ H_1: f(x)=\frac{1}{2}\,e^{-|x|};\quad -\infty<x<\infty. \]
For testing \(H_0\) against \(H_1\), let \(R\) denote the critical region based on the likelihood ratio test having level \(0.05\). Then, for some constant \(c\), the region \(R\) is

Let \(X_1,X_2,\ldots,X_n\ (n>1)\) be a random sample from the following probability density function
\[ f_{\beta}(x)=\begin{cases}\beta e^{-x}(1-e^{-x})^{\beta-1} & \text{if } x>0\\ 0 & \text{otherwise},\end{cases} \]
where \(\beta>0\) is an unknown parameter. For testing the following hypotheses,
\[ H_0:\beta=1 \quad \text{against} \quad H_1:\beta>1, \]
at level \(\alpha\in(0,1)\), which of the following statements is correct?

Let \(X_1, X_2, \ldots, X_5\) be random observations from a continuous distribution. Let \(\theta_p\) be the \(p\)-th population quantile. Consider the following hypotheses
\[ H_0:\theta_{1/2}=2.5 \quad \text{against} \quad H_1:\theta_{1/2}>2.5. \]
Let \(X_{(r)}\) denote the \(r\)-th order statistic of the given observations. Then which of the following is a critical region of a level \(0.05\) test?

Let \(X\) be a single observation from a distribution having a probability density function \(f_\theta\), \(\theta \in \Theta\). For testing \(H_0: \theta = 1\) against \(H_1: \theta \neq 1\), under which of the following options a uniformly most powerful test of level \(0.05\) exists?

Let \(X\) be a single observation from a distribution having probability density function
\[ f_\theta(x)=\begin{cases}\dfrac{2x}{\theta^2}&\text{if }0<x<\theta\\0&\text{otherwise,}\end{cases} \]
where \(\theta\in(0,\infty)\). For testing \(H_0:\theta\leq1\) against \(H_1:\theta>1\), at level of significance \(0.05\), let \(\beta_1\) be the size of the uniformly most powerful test and \(\beta_2\) be the power of the uniformly most powerful test at \(\theta=2\). Then \(10\beta_1+40\beta_2\) equals ______ (answer in integer).