Unbiased and Large-sample Tests - Testing of Hypotheses - Statistics Previous Year Questions

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

Consider the following one-way fixed effects analysis of variance model $Y_{ij} = \mu + \tau_i + \epsilon_{ij}, i = 1,2,3; j = 1,2,3,4;$ where $\epsilon_{ij}$'s are independent and identically distributed $N(0, \sigma^2)$ random variables, $\sigma \in (0, \infty)$ and $\tau_1 + \tau_2 + \tau_3 = 0$. Let $MST$ and $MSE$ denote the mean sum of squares due to treatment and the mean sum of squares due to error, respectively. For testing $H_0: \tau_1 = \tau_2 = \tau_3 = 0$ against $H_1: \tau_i \neq 0$, for some $i = 1,2,3$, consider the test based on the statistic $\frac{MST}{MSE}$. For positive integers $\nu_1$ and $\nu_2$, let $F_{\nu_1, \nu_2}$ be a random variable having the central $F$-distribution with $\nu_1$ and $\nu_2$ degrees of freedom. If the observed value of $\frac{MST}{MSE}$ is given to be 104.45, then the p-value of this test equals

Let \( X \) and \( Y \) be two random variables such that \( p_{11} + p_{10} + p_{01} + p_{00} = 1 \), where \( p_{ij} = P(X = i, Y = j) \), \( i, j = 0, 1 \). Suppose that a realization of a random sample of size \( 60 \) from the joint distribution of \( (X, Y) \) gives \( n_{11} = 10, n_{10} = 20, n_{01} = 20 \) and \( n_{00} = 10 \), where \( n_{ij} \) denotes the frequency of \( (i, j) \) for \( i, j = 0, 1 \). If the chi-square test of independence is used to test \( H_0: p_{ij} = p_{i.} p_{.j} \) for \( i, j = 0, 1 \) against \( H_1: p_{ij} \neq p_{i.} p_{.j} \) for at least one pair \( (i, j) \), where \( p_{i.} = p_{i0} + p_{i1} \) and \( p_{.j} = p_{0j} + p_{1j} \), then which one of the following statements is true?

Let a random sample of size 100 from a normal population with unknown mean \(\mu\) and variance 9 give the sample mean 5.608. Let \(\Phi(\cdot)\) denote the distribution function of the standard normal random variable. If \(\Phi(1.96) = 0.975\), \(\Phi(1.64) = 0.95\) and the uniformly most powerful unbiased test based on sample mean is used to test \(H_0: \mu = 5.02\) against \(H_1: \mu \neq 5.02\), then the p-value equals __________ (round off to 3 decimal places).

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?

Let \(X_1, X_2, \ldots, X_n\) be a random sample of size \(n\ (n \geq 11)\) from a population with continuous and strictly increasing cumulative distribution function \(F(\cdot)\) with an unknown median \(M\). To test \(H_0: M = 10\) against \(H_1: M > 10\) at level \(\alpha\), let the statistic \(T\) denote the number of observations larger than 10. Let \(t_0\) be the observed value of the test statistic \(T\). Consider the test which rejects \(H_0\) if \(T \geq c\). Then the \(p\)-value of the test is