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

We need to decide which hypotheses correctly represent the claim that students at Alyssa's school spend more than \[tex]$195. Let's define the parameter of interest:

- Let $[/tex]\mu[tex]$ be the true mean price of a prom dress at Alyssa's high school.

Given the information, the survey of 8,000 high school students suggests that a typical prom dress cost \$[/tex]195. Alyssa believes that her high school students are spending more than this amount. This belief is what we test against the general population mean.

Since Alyssa is testing whether the mean in her school is greater than \[tex]$195, we set up the hypotheses as follows:

1. The null hypothesis ($[/tex]H_0[tex]$) represents the status quo or the claim to be tested. It assumes that the mean is equal to \$[/tex]195:
[tex]$$
H_0: \mu = 195
$$[/tex]

2. The alternative hypothesis ([tex]$H_a$[/tex]) reflects Alyssa's claim that the school is more fashion conscious, meaning the mean is greater than \[tex]$195:
$[/tex][tex]$
H_a: \mu > 195
$[/tex][tex]$

This corresponds to a one-sided test where we are looking for evidence that the mean price spent is higher than \$[/tex]195.

Thus, the correct pair of hypotheses is:
[tex]$$
H_0: \mu = 195 \quad \text{and} \quad H_a: \mu > 195.
$$[/tex]

This is the option that correctly reflects the claim that the mean is more than \$195.