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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 with a standard deviation of \$[/tex]12.00. Alyssa thinks that her school is more fashion-conscious, so she collected data from 20 people in her high school and found that the mean prom dress price was \$208.00. Which of the following are the correct null and alternative hypotheses?

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 solve this question, we need to determine the correct null and alternative hypotheses based on Alyssa's claim about the mean price of prom dresses at her school.

### Step 1: Understand the Situation
- The average price of a prom dress, according to a survey, is [tex]$195.00 with a standard deviation of $[/tex]12.00.
- Alyssa believes that prom dresses at her school are more expensive, specifically greater than [tex]$195.00.
- She collects data from 20 students and finds an average price of $[/tex]208.00.

### Step 2: Formulate Hypotheses
When setting up hypotheses, the null hypothesis (H₀) often states that there is no effect or no difference, while the alternative hypothesis (Hₐ) is what you're trying to find evidence for.

#### Null Hypothesis (H₀)
- The null hypothesis would be that the mean price of a prom dress at Alyssa's school is equal to the survey's mean, which is [tex]$195. This is a statement of "no change" or "no difference."

\( H_0: \mu = 195 \)

#### Alternative Hypothesis (Hₐ)
- Alyssa's claim is that the prom dresses are more expensive at her school, implying that the average is greater than $[/tex]195.

[tex]\( H_a: \mu > 195 \)[/tex]

### Conclusion
Based on Alyssa's belief and the data she collected, the correct hypotheses are:
- Null Hypothesis: [tex]\( H_0: \mu = 195 \)[/tex]
- Alternative Hypothesis: [tex]\( H_a: \mu > 195 \)[/tex]

Thus, the correct option is:
- [tex]\( H_0: \mu = 195; H_a: \mu > 195 \)[/tex]

This means the survey's average is taken as the baseline, and Alyssa is testing to see if her school's average is statistically greater than that baseline.

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