Question

select two different ways to state that the difference of means of the two populations is negative.

$h_{1}:mu_{1}-mu_{2}>0$

$h_{1}:mu_{1}-mu_{2}<0$

$h_{1}:mu_{1}>mu_{2}$

$h_{1}:mu_{1}

Explanation:

1. For \( H_1: \mu_1 - \mu_2 < 0 \), the expression \( \mu_1 - \mu_2 \) represents the difference of the means of the two populations. If this difference is less than 0, it means the difference of the means is negative.
2. For \( H_1: \mu_1 < \mu_2 \), if we rearrange this inequality (by subtracting \( \mu_2 \) from both sides), we get \( \mu_1 - \mu_2 < 0 \), which also represents that the difference of the means of the two populations is negative.

The other options: \( H_1: \mu_1 - \mu_2 > 0 \) implies a positive difference of means, and \( H_1: \mu_1 > \mu_2 \) (which is equivalent to \( \mu_1 - \mu_2 > 0 \)) also implies a positive difference of means, so they do not represent a negative difference of means.

Answer:

B. \( H_1: \mu_1 - \mu_2 < 0 \)
D. \( H_1: \mu_1 < \mu_2 \)