Lester Hollar is Vice President for Human Resources for a...

Lester Hollar is Vice President for Human Resources for a large manufacturing company. In recent years he has noticed an increase in absenteeism that he thinks is related to the general health of the employees. Four years ago, in an attempt to improve the situation, he began a fitness program in which employees exercise during their lunch hour. To evaluate the program, he selected a random sample of eight participants and found the number of days each was absent in the six months before the exercise program began and in the last six months. Below are the results. At the .05 significance level, can he conclude that the number of absences has declined? Estimate the $p$ -value.

Key Concepts

Significance Level

The significance level, often denoted as alpha (?), is the threshold set by the researcher to determine when to reject the null hypothesis. It represents the probability of committing a Type I error, which is the error of rejecting a true null hypothesis. In this problem, the significance level is set at 0.05, meaning that if the p-value is below 0.05, Lester can conclude that there is statistically significant evidence of a decline in absenteeism following the intervention.

P-Value

The p-value is the probability of observing the data, or something more extreme, if the null hypothesis is true. It quantifies the strength of the evidence against the null hypothesis. In this case, a smaller p-value indicates stronger evidence that the fitness program is associated with a decline in absenteeism. Researchers compare the p-value to a predetermined significance level to decide whether to reject the null hypothesis.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. In the context of this question, it involves setting up a null hypothesis stating that there is no change (or no decline) in absenteeism, and an alternative hypothesis stating that absenteeism has decreased. The goal is to use the sample data to assess whether there is sufficient evidence to reject the null hypothesis in favor of the alternative.

Paired Sample t-Test

A paired sample t-test is used when the data consists of paired observations, such as measurements taken from the same subjects before and after a treatment or intervention. This test compares the mean of the differences between the paired observations to zero, thereby accounting for the natural variations between individuals. In this scenario, each employee’s number of absences before and after the fitness program is compared, making the paired sample t-test an appropriate choice to evaluate the change in absenteeism.

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