Question

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

choose the correct answer below.

a. they are equivalent through algebraic manipulation.

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

c. the values of $\mu_1$ and $\mu_2$ are equivalent in every population. therefore, these hypotheses are equivalent.

d. by definition, the null hypothesis is always equal to 0. therefore, these hypotheses are equivalent.

Explanation:

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

• Option A: Subtract \( \mu_2 \) from both sides of \( \mu_1 = \mu_2 \), resulting in \( \mu_1 - \mu_2 = 0 \). This is algebraic manipulation, so this option is correct.
• Option B: The null hypothesis being "assumed true" does not explain their equivalence.
• Option C: \( \mu_1 \) and \( \mu_2 \) are not necessarily equivalent in every population (the null hypothesis is a claim to test, not a universal truth).
• Option D: The null hypothesis is not "always equal to 0"—it depends on the claim (e.g., \( H_0: \mu_1 = \mu_2 \) is a valid null hypothesis without being 0 in form).

Answer:

A. They are equivalent through algebraic manipulation.