Chapter 9 Testing Hypotheses

9.1 Problems of Testing Hypotheses

  1. Let \(X\) have the exponential distribution with parameter \(\beta\). Suppose that we wish to test the hypotheses: \[\begin{aligned} H_0:& \;\; \beta \ge 1 \\ H_1:& \;\; \beta < 1 \\ \end{aligned}\] Consider the test procedure \(\delta\) that rejects \(H_0\) if \(X\ge1\).
    1. Determine the power function of the test.
    2. Compute the size of the test.
  2. Suppose that the proportion \(p\) of defective items in a large population of items is unknown, and that it is desired to test the following hypotheses: \[\begin{aligned} H_0: & \;\; p = 0.2 \\ H_1: & \;\; p \ne 0.2 \end{aligned}\] Suppose that a random sample of \(n=20\) items is drawn from the population and let \(Y\) denote the number of defective items observed. Consider a test procedure \(\delta\) such that we will fail to reject \(H_0\) if \(Y \in \{2,3,\dots,6\}\).
    1. Determine the value of the power function \(\pi(p|\delta)\) at the points \(p=\{0,0.1,\dots,0.9, 1\}\). Sketch the power function.
    2. Determine the size of the test.
  3. Suppose that \(X_1, \dots, X_n \stackrel{iid}{\sim} N(\mu, 1)\). Suppose that \(\mu_0\) is some specified number and that we are interested in testing: \[\begin{aligned} H_0: & \;\; \mu = \mu_0\\ H_1: & \;\; \mu \ne \mu_0 \end{aligned}\] Finally, suppose that \(n=25\) and consider a test procedure that will reject \(H_0\) if \(\left| \bar{X}_n -\mu_0 \right|\ge c\) for some value of \(c\). Determine the value of \(c\) such that the size of the test is \(0.05\).

9.2 Simple Hypotheses

  1. Suppose that \(X_1, \dots, X_n \stackrel{iid}{\sim} \textrm{Poisson}(\lambda)\) and we are interested in testing \[\begin{aligned} H_0: & \;\; \lambda = \lambda_0 \\ H_1: & \;\; \lambda = \lambda_1 \end{aligned}\] where \(0 < \lambda_0 < \lambda_1\). Show that the Likelihood Ratio Test procedure reduces to the decision rule to reject \(H_0\) if \(\bar{X}>c\) for some constant \(c\).

  2. Suppose that \(X_1, \dots, X_n \stackrel{iid}{\sim} \textrm{Bernoulli}(p)\). Let \(0 < p_0 < p_1 < 1\) and we are interested in testing \[\begin{aligned} H_0: & \;\; p = p_0 \\ H_1: & \;\; p = p_1 \end{aligned}\] Show that the Likelihood Ratio Test procedure reduces to the decision rule to reject \(H_0\) if the sample proportion \(\hat{p}\) is such that \(\hat{p} > c\) for some constant \(c\).

  3. Suppose that \(X_1, \dots, X_n \stackrel{iid}{\sim} \textrm{Exponential}(\beta)\). Let \(0 < \beta_1 < \beta_0\) and we are interested in testing \[\begin{aligned} H_0: & \;\; \beta = \beta_0 \\ H_1: & \;\; \beta = \beta_1 \end{aligned}\] Show that the Likelihood Ratio Test procedure reduces to the decision rule to reject \(H_0\) if the sample mean \(\bar{X}\) is such that \(\bar{X} > c\) for some constant \(c\).