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]

To determine the correct null hypothesis ([tex]$H_0$[/tex]) and alternate hypothesis ([tex]$H_a$[/tex]), we need to consider Alyssa's belief and what we're testing.

1. Understand the Claim:
Alyssa thinks that students at her high school spend more on prom dresses than the general population average of \[tex]$195.

2. Define the Parameters:
- The population mean for the general survey is $\mu = 195$.
- Alyssa is suggesting that the mean for her high school $\mu$ is greater than \$[/tex]195.

3. Set Up the Hypotheses:
- Null Hypothesis ([tex]$H_0$[/tex]): This is a statement of no effect or no difference. It represents a standard or baseline level. In this case, it is that the average price of a prom dress in Alyssa's school is equal to the general population average price of \[tex]$195.
\[
H_0: \mu = 195
\]

- Alternate Hypothesis ($H_a$): This captures Alyssa's belief or claim about the data, which she wants to test. Here, she thinks that the average price is more than \$[/tex]195.
[tex]\[
H_a: \mu > 195
\][/tex]

Given this setup, the correct hypotheses are:
- [tex]$H_0: \mu = 195$[/tex]
- [tex]$H_a: \mu > 195$[/tex]

This problem involves a one-tailed test because Alyssa is specifically interested in finding out if the mean price is greater than \$195, not simply different (which would require a two-tailed test).