Questions: A shareholders' group is lodging a protest against your company. The shareholders group claimed that the mean tenure for a chief executive office (CEO) was at least 9 years. A survey of 69 companies reported in The Wall Street Journal found a sample mean tenure of 7.4 years for CEOs with a standard deviation of s=5.2 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 a hypothesis that can be used to challenge the validity of the claim made by the group, at a significance level of α=0.002. In other words, your claim is that the mean tenure for a CEO was less than 9 years. Your hypotheses are: H0: μ ≥ 9 H1: μ<9 What is the test statistic for this sample? test statistic = -2.5559 What is the p-value for this sample? p-value = 0.0053

A shareholders' group is lodging a protest against your company. The shareholders group claimed that the mean tenure for a chief executive office (CEO) was at least 9 years. A survey of 69 companies reported in The Wall Street Journal found a sample mean tenure of 7.4 years for CEOs with a standard deviation of s=5.2 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 a hypothesis that can be used to challenge the validity of the claim made by the group, at a significance level of α=0.002. In other words, your claim is that the mean tenure for a CEO was less than 9 years. Your hypotheses are: H0: μ ≥ 9 H1: μ<9 What is the test statistic for this sample? test statistic = -2.5559 What is the p-value for this sample? p-value = 0.0053
Transcript text: A shareholders' group is lodging a protest against your company. The shareholders group claimed that the mean tenure for a chief executive office (CEO) was at least 9 years. A survey of 69 companies reported in The Wall Street Journal found a sample mean tenure of 7.4 years for CEOs with a standard deviation of $s=5.2$ 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 a hypothesis that can be used to challenge the validity of the claim made by the group, at a significance level of $\alpha=0.002$. In other words, your claim is that the mean tenure for a CEO was less than 9 years. Your hypotheses are: \[ \begin{array}{l} H_{o}: \mu \geq 9 \\ H_{1}: \mu<9 \end{array} \] What is the test statistic for this sample? test statistic $=$ $-2.5559$ What is the $p$-value for this sample? p-value = 0.0053
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Solution

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Solution Steps

Step 1: Standard Error Calculation

To calculate the standard error \( SE \) of the sample mean, we use the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{5.2}{\sqrt{69}} \approx 0.626 \]

Step 2: Test Statistic Calculation

The test statistic \( Z_{test} \) is calculated using the formula: \[ Z_{test} = \frac{\bar{x} - \mu_0}{SE} = \frac{7.4 - 9}{0.626} \approx -2.5559 \]

Step 3: P-value Calculation

For a left-tailed test, the p-value is determined using the standard normal distribution: \[ P = T(z) \approx 0.0053 \]

Final Answer

The test statistic is \( Z_{test} = -2.5559 \) and the p-value is \( P \approx 0.0053 \).

Thus, the final answers are: \[ \boxed{Z_{test} = -2.5559} \] \[ \boxed{P = 0.0053} \]

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