Type II error for a lower-tail test at true mean 190

Question

An engineer tests H0: \mu = 200 against H1: \mu < 200 with a known \sigma = 30, a sample of n = 36, at the one-sided 5% level. Suppose the true mean is actually \mu = 190. Find the Type II error rate \beta (the probability the test fails to reject even though H0 is false).

Accepted Answer

Standard error: 	ext{SE} = 30/\sqrt{36} = 30/6 = 5. The lower-tail test rejects when \bar x < 200 - z{0.05}\,	ext{SE} = 200 - 1.645(5) = 191.775. A Type II error occurs when the test fails to reject, i.e. \bar x \geq 191.775, even though the true mean is 190. Evaluate that under \mu = 190: \beta = P(\bar x 191.775 \mid \mu = 190) = P\!\left(Z \frac{191.775 - 190}{5}ight) = P(Z 0.355). \beta = 1 - \Phi(0.355) = 1 - 0.6387 \approx 0.36. So the power is 1 - \beta \approx 0.64. Conclusion: with a true drop to 190, the test still misses it about 36% of the time. Takeaway. \beta is the area of …

Type II error for a lower-tail test at true mean 190

Your answer

Discussion

Learn the concepts

Related questions