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 = 195[/tex]

D. [tex]H_0: \mu < 195[/tex]; [tex]H_a: \mu \geq 208[/tex]

We want to test if the average prom dress price in Alyssa’s high school is greater than \[tex]$195. This leads us to set up our hypotheses as follows:

1. The null hypothesis, $[/tex]H_0[tex]$, represents the assumption that there is no difference from the known value. In this case, it states that the mean price is \$[/tex]195. Formally,
[tex]$$H_0: \mu = 195.$$[/tex]

2. The alternative hypothesis, [tex]$H_a$[/tex], expresses Alyssa's claim that the mean price is higher than \[tex]$195. Thus, we have
$[/tex][tex]$H_a: \mu > 195.$[/tex][tex]$

These correspond to the first option:
$[/tex][tex]$H_0: \mu = 195 \quad \text{and} \quad H_a: \mu > 195.$[/tex]$