Testing of Hypotheses - Statistics Previous Year Questions

A \(2^3\) factorial experiment with factors A, B and C is arranged in two blocks of four plots each as follows: (Below (1) denotes the treatment in which A, B and C are at the lower level, ac denotes the treatment in which A and C are at the higher level and B is at the lower level and so on.)

Block 1	(1)	ab	ac	bc
Block 2	a	b	c	abc

The treatment contrast that is confounded with the blocks is

Consider the following two-way fixed effects analysis of variance model \(Y_{ijk} = \mu + \alpha_i + \beta_j + \epsilon_{ijk}, \quad i = 1,2; j = 1,2,3; k = 1,2,3;\), where \(\epsilon_{ijk}\)'s are independently and identically distributed \(N(0, \sigma^2)\) random variables, \(\sigma \in (0, \infty)\), \(\alpha_1 + \alpha_2 = 0\) and \(\beta_1 + \beta_2 + \beta_3 = 0\). Let \(SSE\) denote the sum of squares due to error. For any positive integer \(\nu\) and any \(\alpha \in (0,1)\), let \(\chi^2_{\nu, \alpha}\) denote the \((1-\alpha)\)-th quantile of the central chi-square distribution with \(\nu\) degrees of freedom. Then a 95% confidence interval for \(\sigma^2\) is given by

Let \(X_1, X_2, \ldots, X_n\) be a random sample of size \(n\) (≥ 2) from a \(N(0, \sigma^2)\) distribution. For a given \(\sigma > 0\), let \(f_\sigma\) denote the joint probability density function of \((X_1, X_2, \ldots, X_n)\) and \(S = \{f_\sigma : \sigma > 0\}\). Let \(T_1 = \sum_{i=1}^n X_i^2\) and \(T_2 = \left(\frac{1}{n} \sum_{i=1}^n X_i\right)^2\). For any positive integer \(\nu\) and any \(\alpha \in (0, 1)\), let \(\chi_{\nu, \alpha}^2\) denote the \((1 - \alpha)\)-th quantile of the central chi-square distribution with \(\nu\) degrees of freedom. Consider testing \(H_0 : \sigma = 1\) against \(H_1 : \sigma > 1\) at level \(\alpha\). Then which one of the following statements is true?

A random sample $X_1, X_2, \ldots, X_6$ of size 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 statistic is

Suppose that \( X_1, X_2, \ldots, X_n \) are independent and identically distributed random variables each having probability density function \( f(\cdot) \) and median \( \theta \). We want to test
\[ H_0: \theta = \theta_0 \quad \text{against} \quad H_1: \theta > \theta_0. \]
Consider a test that rejects \( H_0 \) if \( S > c \) for some \( c \) depending on the size of the test, where \( S \) is the cardinality of the set \( \{i: X_i > \theta_0, 1 \leq i \leq n\} \). Then which one of the following statements is true?