College

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]

Answer :

Sure! Let's go through the process of determining the correct null and alternate hypotheses for this scenario.

1. Understanding Hypotheses:
- The null hypothesis (H₀) is a statement that there is no effect, no difference, or no change. It represents the status quo or the standard used for comparison.
- The alternate hypothesis (Hₐ) is what we want to test for. It's the statement that indicates there is an effect, a difference, or a change from the status quo.

2. What We Know:
- The mean price of a prom dress for the general population of high school students (8,000 students) is [tex]$195.00.
- Alyssa believes that students at her school spent more on prom dresses, specifically that they spent more than $[/tex]195.00.

3. Formulating Hypotheses:
- The null hypothesis should reflect that there is no difference from what is generally known. Therefore, the null hypothesis (H₀) would be that the mean price is [tex]$195.00.
- H₀: μ = 195
- Alyssa's belief leads to the alternate hypothesis that the mean price is greater than $[/tex]195.00. She believes students spend more.
- Hₐ: μ > 195

4. Conclusion:
- The correct set of hypotheses for Alyssa’s investigation is:
- Null Hypothesis (H₀): μ = 195
- Alternate Hypothesis (Hₐ): μ > 195

This setup allows Alyssa to statistically test if the average amount spent on prom dresses in her school is greater than $195.00, supporting her claim that her school is more fashion conscious.