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A shareholders' group is lodging a protest against your company. The shareholders group claimed that the mean tenure for a chief executive officer (CEO) was less than 8 years. A survey of 54 companies reported in The Wall Street Journal found a sample mean tenure of 6.5 years for CEOs with a standard deviation of [tex]$s=5.9$[/tex] years (The Wall Street Journal, January 2, 2007). You don't know the population standard deviation but can assume it is normally distributed.

You want to formulate and test the hypothesis made by the group, at a significance level of [tex]$\alpha=0.001$[/tex].

Your hypotheses are:

1. Enter the null hypothesis for this test.
[tex]H_0: \mu \geq 8[/tex]

2. Enter the alternative hypothesis for this test.
[tex]H_a: \mu \ \textless \ 8[/tex]

Is the original claim located in the null or alternative hypothesis?
Select an answer.

Sagot :

GinnyAnsweridn
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.
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