It is desired to test H0: μ = 12 against H1: μ ≠ 12 using α = 0.05. The population in question is uniformlydistributed with a standard deviation of 2.0. A random sample of 100 will be drawn from this population. If μis really equal to 11.9, what is the value of β associated with this test?
A) 0.0790
B) 0.9210
C) 0.4210
D) 0.0395

Question

Quizplus · Accepted Answer

To calculate β (the probability of a Type II error), we first determine the z-score for the actual mean (11.9) under the null hypothesis (μ = 12). Using the standard deviation (σ = 2.0) and sample size (n = 100), the standard error (SE) is σ/√n = 2/√100 = 0.2. The z-score for the difference (12 - 11.9 = 0.1) is 0.1/0.2 = 0.5. For a two-tailed test with α = 0.05, the critical z-values are ±1.96. To find β, we look at the z-table for the probability less than 1.96 - 0.5 = 1.46 and greater than -1.96 - 0.5 = -2.46. The z-table gives us the probabilities corresponding to these z-scores, and the area between them represents 1 - β. The closest value to the calculated area in the options given is 0.0790, indicating the probability of failing to reject H0 when μ = 11.9.

It Is Desired to Test H0: μ = 12 Against