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

Answer :

To tackle this question, we need to identify the correct null hypothesis ([tex]\(H_0\)[/tex]) and alternate hypothesis ([tex]\(H_a\)[/tex]) based on Alyssa's belief and the given information.

Here are the steps to find the hypotheses:

1. Understand the Given Situation:
- The mean price of a prom dress according to the survey is [tex]$195.
- Alyssa believes that students at her school spent more than $[/tex]195 on a prom dress.
- The sample data from her school showed an average price of [tex]$208.

2. Formulate the Hypotheses:
- Null Hypothesis (\(H_0\)): The null hypothesis is a statement of no effect or no difference. It is what we assume to be true until we have evidence to suggest otherwise. In this context, the null hypothesis would state that the mean price of a prom dress for Alyssa's school is the same as the surveyed average. Therefore, we set:
\[
H_0: \mu = 195
\]
This means there is no difference, and the average price of a prom dress at Alyssa's school is $[/tex]195.

- Alternate Hypothesis ([tex]\(H_a\)[/tex]): The alternate hypothesis is what we want to test and is typically the statement Alyssa is trying to prove. Since Alyssa thinks students at her school spent more, the alternate hypothesis should reflect an increase in the mean price:
[tex]\[
H_a: \mu > 195
\][/tex]
This means the average price of a prom dress at Alyssa's school is greater than [tex]$195.

3. Select the Correct Hypotheses Pair:
- From the options given:
- \(H_0: \mu = 195\) and \(H_a: \mu > 195\)

The correct pair of hypotheses based on Alyssa's belief that the average prom dress cost is more than \$[/tex]195 is:

- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 195\)[/tex]
- Alternate Hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu > 195\)[/tex]

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