High School

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 \neq 195 ; H_{a}: \mu = 208[/tex]

C. [tex]H_0: \mu = 195 ; H_{a}: \mu = 195[/tex]

D. [tex]H_0: \mu < 195 ; H_{a}: \mu \geq 208[/tex]

Answer :

To determine the correct null and alternate hypotheses for this problem, let's break down the information:

1. Understanding the Question:
Alyssa believes that students at her school spend more than [tex]$195 on prom dresses. To test her belief, we need to set up hypotheses comparing the mean prices.

2. Setting Up Hypotheses:
- Null Hypothesis (H₀): This is the statement we initially assume to be true or the status quo. In this context, we assume that the average amount spent is equal to $[/tex]195. Therefore, [tex]\( H_0: \mu = 195 \)[/tex].
- Alternative Hypothesis (Hₐ): This represents what Alyssa wants to test or prove. Since she believes students spend more, the alternative hypothesis is that the average spending is greater than [tex]$195. Thus, \( H_a: \mu > 195 \).

3. Reviewing Options:
- \( H_0: \mu = 195 \); \( H_a: \mu > 195 \) (This matches Alyssa's belief.)
- \( H_0: \mu \neq 195 \); \( H_a: \mu = 208 \) (This is inconsistent as it doesn't directly relate to her belief that students spend more than $[/tex]195.)
- [tex]\( H_0: \mu = 195 \)[/tex]; [tex]\( H_a: \mu = 195 \)[/tex] (This isn't a valid test as both hypotheses are the same.)
- [tex]\( H_0: \mu < 195 \)[/tex]; [tex]\( H_a: \mu \geq 208 \)[/tex] (This contradicts the hypothesis test we need, where we are checking if the mean is greater than $195.)

Therefore, the correct hypotheses reflecting Alyssa's claim are:

Null Hypothesis (H₀): [tex]\(\mu = 195\)[/tex]
Alternative Hypothesis (Hₐ): [tex]\(\mu > 195\)[/tex]