Answer :
In hypothesis testing, we formulate two statements: the null hypothesis (H0) and the alternative hypothesis (Ha). These hypotheses are designed to test a claim about a population parameter, often the mean ([tex]\mu[/tex]).
Given the statement [tex]\mu = 99[/tex], we want to write its complement and determine which hypothesis is the null and which is the alternative.
Option A:
- [tex]H0: \mu \geq 99[/tex] (claim)
- [tex]Ha: \mu < 99[/tex]
This option suggests that the claim is that the mean is at least 99, and the alternative hypothesis is that it is less than 99.
Option B:
- [tex]H0: \mu \leq 99[/tex]
- [tex]Ha: \mu > 99[/tex] (claim)
Here, the claim suggests that the mean is greater than 99, and the null hypothesis states it is 99 or less.
Option C:
- [tex]H0: \mu = 99[/tex] (claim)
- [tex]Ha: \mu \neq 99[/tex]
In this scenario, the claim is that the mean is exactly 99, and the alternative is that it is not 99.
Option D:
- [tex]H0: \mu \neq 99[/tex]
- [tex]Ha: \mu = 99[/tex] (claim)
This suggests that the claim is the mean equals 99, assuming the mean is not 99 is the null hypothesis.
The question specifically asks for the complement of [tex]\mu = 99[/tex], which means testing if the mean is any value other than 99. The alternative is [tex]\mu \neq 99[/tex]. Thus, the correct choice in this case is Option C.
