Chapter 10: Problem 61
Let \(\mu\) denote the true average lifetime (in hours) for a certain type of battery under controlled laboratory conditions. A test of \(H_{0}: \mu=10\) versus \(H_{a}\) : \(\mu<10\) will be based on a sample of size \(36 .\) Suppose that \(\sigma\) is known to be 0.6 , from which \(\sigma_{\bar{x}}=.1\). The appropriate test statistic is then $$ z=\frac{\bar{x}-10}{0.1} $$ a. What is \(\alpha\) for the test procedure that rejects \(H_{0}\) if \(z \leq-1.28 ?\) b. If the test procedure of Part (a) is used, calculate \(\beta\) when \(\mu=9.8\), and interpret this error probability. c. Without doing any calculation, explain how \(\beta\) when \(\mu=9.5\) compares to \(\beta\) when \(\mu=9.8\). Then check your assertion by computing \(\beta\) when \(\mu=9.5\). d. What is the power of the test when \(\mu=9.8\) ? when \(\mu=9.5 ?\)
Short Answer
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Key Concepts
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