A recent survey of 8,000 high school students found that the mean price of a prom dress was [tex]\$195.00[/tex] with a standard deviation of [tex]\$12.00[/tex]. Alyssa thinks that her school is more fashion-conscious and that students spent more than [tex]\$195.00[/tex]. She collected data from 20 people in her high school and found that the average price spent on a prom dress was [tex]\$208.00[/tex].

Which of the following are the correct null hypothesis and alternate hypothesis?

A. [tex]H_0: \mu = 195[/tex]; [tex]H_a: \mu > 195[/tex]
B. [tex]H_0: \mu \neq 195[/tex]; [tex]H_a: \mu = 208[/tex]
C. [tex]H_0: \mu = 195[/tex]; [tex]H_a: \mu \neq 195[/tex]
D. [tex]H_0: \mu < 195[/tex]; [tex]H_a: \mu \geq 208[/tex]

To solve the problem, we start by understanding Alyssa's claim. The survey of 8,000 high school students provided a mean prom dress price of \[tex]$195 with a standard deviation of \$[/tex]12. Alyssa believes that her school is more fashion conscious, meaning that students probably spend more than \[tex]$195 on their prom dresses.

In hypothesis testing, we set up the hypotheses as follows:

1. The null hypothesis ($[/tex]H_0[tex]$) represents the status quo or no-effect scenario. In this case, it states that the mean prom dress price is the same as the survey result. Thus,
$[/tex][tex]$ H_0: \mu = 195. $[/tex][tex]$

2. The alternative hypothesis ($[/tex]H_a[tex]$) reflects the claim that we are testing. Since Alyssa's claim is that students at her school spend more than \$[/tex]195, the alternative hypothesis should state that the mean is greater than 195. Thus,
[tex]$$ H_a: \mu > 195. $$[/tex]

This setup clearly matches the option:
[tex]$$ H_0: \mu = 195 \quad ; \quad H_a: \mu > 195. $$[/tex]

Therefore, the correct null and alternate hypotheses are:
- Null hypothesis: [tex]$$ H_0: \mu = 195 $$[/tex]
- Alternative hypothesis: [tex]$$ H_a: \mu > 195 $$[/tex]

This decision corresponds to the first option.