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 \textgreater 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 \textless 195 ; H_a: \mu \geq 208[/tex]

Answer :

Sure! Let's break down how to determine the correct null and alternate hypotheses for this situation.

1. Understanding the Context: Alyssa has a hypothesis that students at her school spend more on prom dresses than the average price of [tex]$195 reported from a survey of 8,000 high school students.

2. Null Hypothesis (\(H_0\)): The null hypothesis is a statement that there is no effect or no difference, and it often includes a statement of equality. In this context, the null hypothesis would be that the mean price of a prom dress for Alyssa's school is $[/tex]195. This can be written as:
[tex]\[
H_0: \mu = 195
\][/tex]

3. Alternate Hypothesis ([tex]\(H_a\)[/tex]): The alternate hypothesis is what you are trying to find evidence for. Alyssa believes students at her school spent more than the average [tex]$195. Therefore, the alternate hypothesis should reflect that the mean is greater than 195. This can be expressed as:
\[
H_a: \mu > 195
\]

4. Choosing the Correct Option: By looking at the statements and goals behind the hypotheses, the correct set of hypotheses is:
- Null Hypothesis: \(H_0: \mu = 195\)
- Alternate Hypothesis: \(H_a: \mu > 195\)

This selection reflects that Alyssa wants to test if the mean price is greater than the average reported price of $[/tex]195.

Therefore, the statement [tex]\(H_0: \mu=195 ; H_a: \mu>195\)[/tex] is the correct choice.