List of top Testing of Hypotheses Questions

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 ________?