Sure, let's go through each part of the problem step-by-step:
1) What is the null hypothesis [tex]$H_0$[/tex]?
The null hypothesis, denoted by [tex]$H_0$[/tex], is a statement we are trying to test. It usually specifies that there is no effect or no difference, and it is the hypothesis that the population parameter (in this case, the population mean [tex]$\mu$[/tex]) is equal to a certain value.
Given the possible answers:
- [tex]$\bar{x}=79.4$[/tex]
- [tex]$\mu=83$[/tex]
- [tex]$t=-1.776$[/tex]
The correct null hypothesis is [tex]$\mu = 83$[/tex] because it states that the population mean [tex]$\mu$[/tex] is equal to 83.
Correct answer: [tex]$\mu = 83$[/tex]
2) What is the alternative hypothesis [tex]$H_a$[/tex]?
The alternative hypothesis, denoted by [tex]$H_a$[/tex], is a statement that indicates the presence of an effect or difference. It is what we aim to provide evidence for, and it is the opposite of the null hypothesis.
Given the possible answers:
- [tex]$\mu \neq 83$[/tex]
- [tex]$\mu < 83$[/tex]
- [tex]$\mu > 83$[/tex]
Since we are testing whether the population mean is actually lower than the hypothesized mean of 83, the alternative hypothesis would be [tex]$\mu < 83$[/tex].
Correct answer: [tex]$\mu < 83$[/tex]
3) What is the [tex]$p$[/tex]-value?
The [tex]$p$[/tex]-value is the probability of obtaining test results at least as extreme as the observed results under the assumption that the null hypothesis is true.
Given the possible answers:
- 0.082
- 0.041
The correct [tex]$p-value$[/tex] obtained is 0.041.
Correct answer: 0.041
4) What is the conclusion for the [tex]$t$[/tex]-test?
The conclusion is determined by comparing the [tex]$p$[/tex]-value to a significance level (commonly [tex]$\alpha = 0.05$[/tex]). If the [tex]$p$[/tex]-value is less than [tex]$\alpha$[/tex], you reject the null hypothesis; otherwise, you fail to reject it.
Given the [tex]$p$[/tex]-value of 0.041:
Since 0.041 is less than 0.05, we reject the null hypothesis [tex]$H_0$[/tex].
Correct answer: Reject [tex]$H_0$[/tex]
