High School

A recent survey of 8,000 high school students found that the mean price of a prom dress was [tex]\$195[/tex]. Which of the following are the correct null hypothesis and alternative hypothesis?

A. [tex] H_0: \mu = 195 \quad H_a: \mu \neq 208 [/tex]

B. [tex] H_0: \mu = 195 \quad H_a: \mu \neq 195 [/tex]

C. [tex] H_0: \mu = 195 \quad H_a: \mu \geq 208 [/tex]

Answer :

In hypothesis testing for a population mean, we start by setting up the null and alternative hypotheses. The standard procedure is:

1. The null hypothesis, denoted by [tex]$H_0$[/tex], states that there is no difference or effect. In this case it asserts that the population mean is equal to the claimed value. Therefore, we write:
[tex]$$
H_0: \mu = 19500.
$$[/tex]

2. The alternative hypothesis, denoted by [tex]$H_1$[/tex], is a statement that contradicts the null hypothesis. Often, when detecting any difference regardless of direction, a two-sided (or not-equal-to) alternative is used. This is expressed as:
[tex]$$
H_1: \mu \neq 19500.
$$[/tex]

This setup represents a two-tailed test where any significant deviation from [tex]$\$[/tex]19500[tex]$ in either direction would lead us to question the claim that the population mean is $[/tex]\[tex]$19500$[/tex].

Thus, the correct pair of hypotheses is:
- Null hypothesis: [tex]$\mu = 19500$[/tex]
- Alternative hypothesis: [tex]$\mu \neq 19500$[/tex]

This corresponds to option 1 from the given choices.

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