Discussion
The probability density function of the random variable \(X\) is
\[ f(x) = \begin{cases} \frac{1}{\lambda} e^{-x/\lambda}, & x \gt 0 \\ 0, & x \le 0, \end{cases} \]
where \(\lambda \gt 0\). For testing the hypothesis \(H_{0} : \lambda = 3\) against \(H_{A} : \lambda = 5\), a test is given as "Reject \(H_{0}\) if \(X \ge 4.5\)". The probability of type I error and power of this test are, respectively,
Answer: D
The type I error is 0.2231 and the power is 0.4066, so the answer is 0.2231 and 0.4066.
For this exponential distribution with mean \(\lambda\), \(P(X \ge c) = e^{-c/\lambda}\).
\[ \begin{aligned} \alpha &= P(X \ge 4.5 \mid \lambda = 3) = e^{-1.5} = 0.2231 \\ 1 - \beta &= P(X \ge 4.5 \mid \lambda = 5) = e^{-0.9} = 0.4066 \end{aligned} \]
The type I error is the probability of rejecting \(H_{0}\) when it is true, and the power is the probability of rejecting it when \(H_{A}\) is true. 0.1353 is \(e^{-2}\) and 0.4493 is \(e^{-0.8}\); neither corresponds to the threshold 4.5 with means 3 and 5, and the pairs 0.1827/0.379 and 0.2021/0.4493 do not either.
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