Question

The following model is a simplified version of the multiple regression model used by Biddle and Hamermesh (1990) to study the tradeoff between time spent sleeping and working and to look at other factors affecting sleep: $$\text { sleep }=\beta_{0}+\beta_{1} \text { totwr } k+\beta_{2} e d u c+\beta_{3} a g e+u$$ where sleep and totwrk (total work) are measured in minutes per week and educ and age are measured in years. (See also Computer Exercise \(\mathrm{C} 3\) in Chapter \(2 .\) ) i. If adults trade off sleep for work, what is the sign of \(\beta_{1} ?\) ii. What signs do you think \(\beta_{2}\) and \(\beta_{3}\) will have? iii. Using the data in SLEEP75, the estimated equation is $$\begin{aligned} \widehat{\text { sleep }} &=3,638.25-.148 \text { totwrk }-11.13 \text { educ }+2.20 \text { age } \\ n &=706, R^{2}=.113 . \end{aligned}$$ If someone works five more hours per week, by how many minutes is sleep predicted to fall? Is this a large tradeoff? iv. Discuss the sign and magnitude of the estimated coefficient on \(e d u c .\) v. Would you say totwrk, educ, and age explain much of the variation in sleep? What other factors might affect the time spent sleeping? Are these likely to be correlated with totwrk?

Short answer

i. β1 is negative. ii. β2 likely negative, β3 positive. iii. Sleep falls by 44.4 minutes; moderate tradeoff. iv. Educ's coefficient is -11.13. v. Factors explain little; health, stress might also affect sleep, possibly correlated with totwrk.

Step-by-step solution

Identifying the Sign of β1

To determine the sign of \(\beta_1\) in the regression model, consider the relationship between sleep and total work, \(\text{totwrk}\). If adults trade off sleep for work, an increase in work hours would lead to a decrease in sleep hours. Therefore, \(\beta_1\) should be negative, indicating an inverse relationship.

Predicting Signs of β2 and β3

For \(\beta_2\), consider the effect of education \(\text{educ}\) on sleep: more education might involve a more demanding schedule, possibly reducing sleep, so \(\beta_2\) could be negative. For \(\beta_3\), age \(\text{age}\) may positively affect sleep as older adults might require or prioritize more sleep, suggesting \(\beta_3\) could be positive.

Calculating the Effect of Increased Work

For a five-hour increase in work per week, convert hours to minutes (5 hours = 300 minutes). Then, multiply this by the coefficient of \(\text{totwrk}\) from the estimated equation: \[ -0.148 \times 300 = -44.4 \]This indicates sleep is predicted to decrease by 44.4 minutes per week, a relatively moderate tradeoff.

Analyzing the Coefficient of Educ

The estimated coefficient of \(\text{educ}\) is \(-11.13\), implying that each additional year of education is associated with 11.13 fewer minutes of sleep per week. This reflects a small, but consistent negative relationship between education and sleep.

Assessing Explanation of Sleep Variation

The \(R^2\) value of 0.113 suggests that \(\text{totwrk}\), \(\text{educ}\), and \(\text{age}\) explain only about 11.3% of the variation in sleep. Other factors that might affect sleep include health, lifestyle, stress, and family obligations, which may or may not correlate with the variables in the model. Work stress or economic pressure could be correlated with \(\text{totwrk}\).

Key concepts

Tradeoff Between Sleep and Work

In the study of sleep patterns, one critical aspect examined is the tradeoff between sleep and work. This tradeoff is represented by the coefficient of the variable for total work hours, denoted as \(\beta_1\) in our regression model. When individuals increase their work hours, they often do so at the expense of their sleep. Consequently, \(\beta_1\) is expected to be negative, as shown in the model: \[-0.148 \times \text{totwrk}\]. This negative coefficient indicates that for every additional hour spent on work, the corresponding sleep time decreases.

In practical terms, if we consider a scenario where someone works five more hours each week, translating to 300 minutes, the equation predicts a decrease in sleep by approximately 44.4 minutes. While this might seem moderate on a weekly basis, over time, it can significantly impact overall well-being. This tradeoff highlights a crucial balancing act many adults face, emphasizing the importance of recognizing work-life balance in today's hectic lifestyle.

Influences of Education on Sleep

Education also plays a significant role in the variation of sleep patterns, as reflected in the coefficient \(\beta_2\) in our regression equation. Education might demand more time for activities such as studying and commuting to educational institutions. Often, highly educated individuals possess demanding careers, which can also cut into their sleep time. This contributes to the negative coefficient of \(-11.13\) identified for the education variable in the estimated equation.

This effect reveals that each additional year of education tends to decrease sleep by approximately 11.13 minutes per week. Although this appears to be a modest decrease, it stresses how the pursuit of educational and career advancements could cumulatively reduce sleep. Understanding this trend can help individuals devise strategies to optimize their schedules, ensuring necessary sleep without compromising educational achievements.

Factors Affecting Sleep Variation

Beyond work and education, numerous other factors can influence sleep variation, captured by the \(R^2\) statistic, which indicates how much of the variability in sleep is explained by the model. Here, the \(R^2\) value of 0.113 suggests that total work, education, and age account for only 11.3% of sleep variation. This implies the presence of several unaccounted influences on sleep.

Key factors not reflected in the regression model might include:

• Health conditions, such as sleep disorders or chronic illnesses.
• Lifestyle choices, including diet, exercise, and screen time before bed.
• Stress and mental health issues, which can disrupt sleep patterns.
• Family obligations, such as caregiving responsibilities or parenting.

Understanding these factors helps create a comprehensive picture of what affects sleep, underscoring the complex interplay between lifestyle, physiological, and psychological components that extend beyond work and education.