Final answer:
The question asks to calculate the power of a hypothesis test concerning a population mean with known standard deviation, given a specific alternative mean value. It involves finding the z-score at a significance level of 0.05 for a left-tailed test and comparing it to the test statistic for the given alternative mean.
Step-by-step explanation:
The question involves testing a hypothesis about the population mean (μ) given a known population standard deviation (σ) using a sample of size 100.
When testing the hypothesis H0: μ ≥ 30 against Ha: μ < 30 using a significance level (alpha α) of 0.05, we want to determine the power of the test when the true population mean (μ) is 28. The power of a test is the probability of correctly rejecting a false null hypothesis, which in this case refers to detecting a true mean of 28 when the null states it is at least 30.
To calculate the power, first find the critical value (z-score) corresponding to a 0.05 significance level for a left-tailed test. Then calculate the test statistic for μ = 28 given the standard error, which is σ divided by the square root of the sample size. Compare this test statistic to the critical value to find the power of the test. The area to the left of this test statistic in the standard normal distribution gives the power of rejecting the null hypothesis when the actual mean is 28.