A shareholders' group, in lodging a protest, claimed that...

A shareholders' group, in lodging a protest, claimed that the mean tenure for a chief exective office (CEO) was ar least nine years. A survey of companies reported in The Wall Sirect Joumal found a sample mean tenure of $\bar{x}-7.27$ years for CEOs with a standard deviation of $s-6.38$ years (The Wall Sireet Journal, January 2, 2007). a. Formulate hypotheses that can be used to challenge the validity of the claim made by the sharcholders' group. b. Assume 85 companies were included in the sample. What is the p-value for your hypochesis test? c. At $a=.01$, what is your conclusion?

A shareholders' group, in lodging a protest, claimed that the mean tenure for a chief exective office (CEO) was ar least nine years. A survey of companies reported in The Wall Sirect Joumal found a sample mean tenure of $\bar{x}-7.27$ years for CEOs with a standard deviation of $s-6.38$ years (The Wall Sireet Journal, January 2, 2007).
a. Formulate hypotheses that can be used to challenge the validity of the claim made by the sharcholders' group.
b. Assume 85 companies were included in the sample. What is the p-value for your hypochesis test?
c. At $a=.01$, what is your conclusion?

Key Concepts

p-value

The p-value is the probability of obtaining a test result at least as extreme as the one observed, assuming that the null hypothesis is true. It provides a measure of evidence against the null hypothesis; a lower p-value indicates stronger evidence that the null hypothesis may be false. The p-value is compared to a predetermined significance level to make a decision regarding the null hypothesis.

Significance Level

The significance level (denoted as ?) is a threshold set by the researcher before testing, which determines the criteria for rejecting the null hypothesis. It represents the probability of committing a Type I error, i.e., rejecting the null hypothesis when it is actually true. Comparing the p-value to the significance level helps in deciding whether the observed data provide sufficient evidence to reject the null hypothesis.

Null and Alternative Hypotheses

The null hypothesis (H0) typically represents the status quo or a baseline claim (e.g., the mean is at least a certain value), while the alternative hypothesis (Ha) reflects the claim that one seeks to prove (e.g., the mean is lower than that value). Formulating these hypotheses correctly is a critical step, as subsequent analysis—through test statistics and p-values—relies on this setup to assess the evidence against the null hypothesis.

Hypothesis Testing

Hypothesis testing is a fundamental statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. It involves comparing the observed data against what would be expected under a specific assumption, allowing analysts to determine if the data support or contradict a stated claim.

Test Statistic and t-Distribution

When the population standard deviation is unknown and the sample size is moderate or small, the t-distribution is used to compute the test statistic. The test statistic measures how far the sample statistic (such as the sample mean) deviates from the hypothesized population parameter, scaled by the estimated standard error. This quantification aids in determining the probability of observing a result as extreme or more extreme than the sample result if the null hypothesis were true.

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