Which of the following represents the null and alternative hypotheses for testing whether the population mean equals 50 against a one-sided alternative greater than 50?

• A. H0: μ ≤ 50 vs Ha: μ > 50
• B. H0: μ ≠ 50 vs Ha: μ = 50
• C. H0: μ > 50 vs Ha: μ ≤ 50
• D. H0: μ = 50 vs Ha: μ > 50

Answer: (D)

When you test a claim about a population mean against a one-sided alternative, you set up the null to specify the exact value you're testing against, and the alternative to reflect the direction of interest. If the goal is to test whether the true mean equals 50 against the idea that it is greater than 50, the natural setup is:

H0: μ = 50 versus Ha: μ > 50.

This matches the stated objective: the null asserts the mean is exactly 50, and the alternative captures the interest in it being larger. Using equality in the null also lets you use the known sampling distribution under μ = 50 to decide whether to reject.

The other formulations don’t fit the prompt. Setting the null as μ ≤ 50 would address a different framing (testing if the mean exceeds 50, but with a different null boundary). Reversing to Ha: μ = 50 would make the alternative a single exact value, not a one-sided increase. A null of μ > 50 would invert the tested claim and imply a different research question. The chosen setup directly aligns with testing whether the mean is greater than 50 while treating 50 as the exact null value.