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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 :

We begin by identifying the claim made by Alyssa. She believes that her school is more fashion conscious and that the average spending on prom dresses is greater than \[tex]$195. In other words, her claim is that the true mean $[/tex]\mu[tex]$ is greater than 195, which can be written as

$[/tex][tex]$
H_a: \mu > 195.
$[/tex][tex]$

In hypothesis testing, the null hypothesis always represents the status quo or a statement of equality. Since the population mean based on the survey is given as $[/tex]\[tex]$195.00$[/tex], the null hypothesis is

[tex]$$
H_0: \mu = 195.
$$[/tex]

Thus, the correct pair of hypotheses is

[tex]$$
H_0: \mu = 195 \quad \text{and} \quad H_a: \mu > 195.
$$[/tex]

This corresponds to option 1.

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