Answer :
To determine the correct null hypothesis and alternate hypothesis, we need to understand what a hypothesis test is and what Alyssa's specific claim is about the situation.
### Key Points:
1. Null Hypothesis ([tex]\(H_0\)[/tex]): This is a statement of no change or no effect. It is what we assume to be true until we have evidence against it. In this context, [tex]\(H_0\)[/tex] would be that the mean price of a prom dress is still [tex]$195.
2. Alternate Hypothesis (\(H_a\)): This is what we want to prove or verify. It's the statement we are testing against the null. Alyssa believes that students at her school spend more than the mean of $[/tex]195 on prom dresses.
### Given Hypothesis Options:
- [tex]\(H_0: \mu = 195 ; H_a: \mu > 195\)[/tex]: This suggests Alyssa's school spends more, which aligns with her belief.
- [tex]\(H_0: \mu = 195 ; H_a: \mu = 208\)[/tex]: This would mean that the average must be exactly 208, which doesn't match a belief of spending more in general.
- [tex]\(H_0: \mu = 195 ; H_a: \mu \neq 195\)[/tex]: This is a two-tailed test, used when looking for any change (either more or less). This doesn't fit Alyssa’s specific claim of spending more.
- [tex]\(H_0: \mu < 195 ; H_a: \mu \geq 208\)[/tex]: This is incorrectly set up, as the null suggests less, not equal, to 195, and doesn’t align with the scenario presented.
### Conclusion:
The correct hypotheses, given Alyssa believes that students at her school spend more than $195, are:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 195\)[/tex]
- Alternate Hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu > 195\)[/tex]
Thus, the correct choice is:
- [tex]\(H_0: \mu = 195 ; H_a: \mu > 195\)[/tex]
### Key Points:
1. Null Hypothesis ([tex]\(H_0\)[/tex]): This is a statement of no change or no effect. It is what we assume to be true until we have evidence against it. In this context, [tex]\(H_0\)[/tex] would be that the mean price of a prom dress is still [tex]$195.
2. Alternate Hypothesis (\(H_a\)): This is what we want to prove or verify. It's the statement we are testing against the null. Alyssa believes that students at her school spend more than the mean of $[/tex]195 on prom dresses.
### Given Hypothesis Options:
- [tex]\(H_0: \mu = 195 ; H_a: \mu > 195\)[/tex]: This suggests Alyssa's school spends more, which aligns with her belief.
- [tex]\(H_0: \mu = 195 ; H_a: \mu = 208\)[/tex]: This would mean that the average must be exactly 208, which doesn't match a belief of spending more in general.
- [tex]\(H_0: \mu = 195 ; H_a: \mu \neq 195\)[/tex]: This is a two-tailed test, used when looking for any change (either more or less). This doesn't fit Alyssa’s specific claim of spending more.
- [tex]\(H_0: \mu < 195 ; H_a: \mu \geq 208\)[/tex]: This is incorrectly set up, as the null suggests less, not equal, to 195, and doesn’t align with the scenario presented.
### Conclusion:
The correct hypotheses, given Alyssa believes that students at her school spend more than $195, are:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 195\)[/tex]
- Alternate Hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu > 195\)[/tex]
Thus, the correct choice is:
- [tex]\(H_0: \mu = 195 ; H_a: \mu > 195\)[/tex]
