Question

explain why the null hypothesis $h_0: \mu_1 \geq \mu_2$ is equivalent to the null hypothesis $h_0: \mu_1 - \mu_2 \geq 0$.

choose the correct answer below.

a. they are equivalent through algebraic manipulation.

b. by definition, the null hypothesis is always greater than 0. therefore, these hypotheses are equivalent.

c. they are equivalent because the null hypothesis, $h_0$, is always assumed to be true.

d. the value of $\mu_1$ is larger than $\mu_2$ in every population. therefore, these hypotheses are equivalent.

Explanation:

To determine why \( H_0: \mu_1 \geq \mu_2 \) is equivalent to \( H_0: \mu_1 - \mu_2 \geq 0 \), we analyze the options:

• Option A: Subtract \( \mu_2 \) from both sides of \( \mu_1 \geq \mu_2 \), resulting in \( \mu_1 - \mu_2 \geq 0 \). This is algebraic manipulation.
• Option B: The null hypothesis is not "always greater than 0"—it depends on the context, so this is incorrect.
• Option C: The null hypothesis being "assumed true" does not explain their equivalence; it refers to hypothesis testing logic, not algebraic equivalence.
• Option D: There is no guarantee \( \mu_1 > \mu_2 \) in every population, so this is false.

Answer:

A. They are equivalent through algebraic manipulation.