Let's start by understanding some basic concepts about hypothesis testing.
When you perform a hypothesis test, you start with two hypotheses:
- Null Hypothesis ([tex]$H_0$[/tex]): This is a statement that there is no effect or no difference. It is the hypothesis that you aim to test against. In this case, the null hypothesis is that the mean age of bus drivers in Chicago is [tex]$\mu=50.3$[/tex] years.
- Alternative Hypothesis ([tex]$H_1$[/tex] or [tex]$H_a$[/tex]): This is the hypothesis that there is an effect or a difference. It is what you would conclude if you reject the null hypothesis.
The outcome of the hypothesis test can be:
- Reject the null hypothesis ([tex]$H_0$[/tex]): This means that there is sufficient evidence to support the alternative hypothesis.
- Fail to reject the null hypothesis ([tex]$H_0$[/tex]): This means that there is not sufficient evidence to support the alternative hypothesis.
Given these concepts, let's interpret a decision that fails to reject the null hypothesis:
If the decision is to fail to reject the null hypothesis ([tex]$H_0: \mu = 50.3$[/tex]), it means that there is not sufficient evidence to conclude that the mean age of bus drivers in Chicago is different from 50.3 years. In simpler terms, we don't have enough evidence to reject the claim that the mean age is 50.3 years; thus, we assume it to be true for the purpose of the hypothesis test.
Therefore, the correct interpretation is:
D. There is not sufficient evidence to reject the claim [tex]$\mu=50.3$[/tex].
