Question

Suppose we are about to sample 100 observations from a normally distributed population where it is known that σ = 20, but µ is unknown. We intend to test H0: µ = 30 against Ha: µ < 30 at α = 0.05.

(a) What values of the sample mean would lead to a rejection of the null hypothesis?

Answer 1

The rejection of the null hypothesis is based on the sample mean falling into a certain range of values. In this case, any sample mean below 28.71 would lead to a rejection of the null hypothesis at α = 0.05.

Step-by-step explanation:

The rejection of the null hypothesis in a hypothesis test is based on the sample mean falling into a certain range of values. In this case, we are testing the hypothesis H0: µ = 30 against Ha: µ < 30 at α = 0.05. Thus, we are looking for values of the sample mean that are significantly smaller than 30.

To determine this range, we calculate the z-score corresponding to an α of 0.05. Using the standard normal distribution table, we find that the z-score for an α of 0.05 is approximately -1.645. We can then calculate the lower bound of the sample mean using the formula:

Lower Bound = µ + (z-score * σ/√n)

Plugging in the known values of σ = 20, α = 0.05, and n = 100, we get:

Lower Bound = 30 + (-1.645) * 20/√100

Simplifying the expression, we find that the lower bound of the sample mean is approximately 28.71. Therefore, any sample mean below 28.71 would lead to a rejection of the null hypothesis at α = 0.05.