Question

Research carried out to investigate the relationship between smoking status of workers and short-term absenteeism rate (hr/mo) yielded the accompanying summary information ("Work-Related Consequences of Smoking Cessation," Academy of Management Journal [1989]: \(606-621) .\) In addition, \(F=2.56\). Construct an ANOVA table, and then state and test the appropriate hypotheses using a .01 significance level. $$\begin{array}{lrl} \text { Status } & \begin{array}{l} \text { Sample } \\ \text { Size } \end{array} & \begin{array}{l} \text { Sample } \\ \text { Mean } \end{array} \\ \hline \text { Continuous smoker } & 96 & 2.15 \\ \text { Recent ex-smoker } & 34 & 2.21 \\ \text { Long-term ex-smoker } & 86 & 1.47 \\ \text { Never smoked } & 206 & 1.69 \end{array}$$

Short answer

After calculating the Total Sum of Squares (TSS), Within-group Sum of Squares (WSS) and Between-group Sum of squares (BSS), we find the ANOVA table parameters. Then, by conducting the ANOVA test using the given F-statistic and significance level of 0.01, we can test the null hypothesis that all group means are equal. If F-statistic > critical value, we reject the null hypothesis.

Step-by-step solution

Calculate the Total Sum of Squares (TSS)

First, calculate the total sum of squares. This involves calculating the sum of the squared deviations of each observation from the overall mean. In this case, the overall mean is the average absenteeism rate across all workers, regardless of smoking status. Multiply the number of members in each group by the squared difference between the group mean and the overall mean, and sum these values for all groups. We denote the overall mean as \(\mu\), and the mean for group i as \(\mu_{i}\). Using given values the calculation would be as follows: \[TSS = 96 \times (\mu - 2.15)^2 + 34 \times (\mu - 2.21)^2 + 86 \times (\mu - 1.47)^2 + 206 \times (\mu - 1.69)^2 \]

Calculate the Within-Group Sum of Squares (WSS)

This is the sum of the squared deviations of each observation from its group mean, calculated separately for each group and then summed across all groups. This value can be calculated using the given F-statistic and the number of groups (k) and observations (N). The formula is: \[WSS = \frac{TSS}{F-statistic + (k-1)}\] where F = 2.56 (given), k is the number of smoker categories (4) and TSS calculated from previous step.

Calculate the Between-Group Sum of Squares (BSS)

This is the difference between the total sum of squares and the within-group sum of squares. Mathematically, it is expressed as: \[BSS = TSS - WSS\] Taking TSS and WSS values from Step 1 and 2.

Conduct the ANOVA test

A significant F-statistic implies that there is a significant variation between the groups. Here, given the significance level of 0.01, we are testing the null hypothesis that all group means are equal against the alternative hypothesis that at least one group mean is different. If the calculated F-statistic is greater than the critical value in the F-distribution, we reject the null hypothesis.