Sure, let's walk through the detailed solution step by step for this hypothesis test.
### Step 1: Understand the claims and hypotheses
The shareholders' group claims that the mean tenure for a CEO is less than 8 years. This is our alternative hypothesis because it suggests a change from the status quo.
### Step 2: Set up the hypotheses
#### Null Hypothesis ([tex]\(H_0\)[/tex]):
The null hypothesis is a statement of no change or no effect. In this scenario, the null hypothesis claims that the mean tenure for a CEO is at least 8 years:
[tex]\[ H_0: \mu \geq 8 \][/tex]
#### Alternative Hypothesis ([tex]\(H_a\)[/tex]):
The alternative hypothesis is what we are trying to find evidence for. According to the shareholders' claim, we formulate the alternative hypothesis as follows:
[tex]\[ H_a: \mu < 8 \][/tex]
### Step 3: Determine the significance level and identify the original claim
The problem states that the significance level ([tex]\(\alpha\)[/tex]) is 0.001, which means there is a 0.1% chance of rejecting the null hypothesis when it is actually true.
The original claim by the shareholders' group is that the mean tenure for a CEO is less than 8 years, which aligns with the alternative hypothesis.
### Step 4: Summary of the solution
Given the information:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\text{mean tenure} \geq 8\)[/tex]
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\text{mean tenure} < 8\)[/tex]
- The original claim is in the alternative hypothesis.
### Final Answer:
- The null hypothesis for this test: [tex]\[ H_0: \text{mean tenure} \geq 8 \][/tex]
- The alternative hypothesis for this test: [tex]\[ H_a: \text{mean tenure} < 8 \][/tex]
- The original claim is located in the alternative hypothesis.
This completes the formulation and identification of hypotheses for the hypothesis test.
