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:
sleep=B_0+B_1 totwork+B_2 educ+B_3 age+u
Where sleep and totwork(total work) are measured in minutes per week and educ and age are measured in years.
If adults trade off sleep for work what is the sign of B_1?
What signs do you think of B_2and B_3?
Suppose the estimated equation is;
(sleep) ̂=3638.25-0.148totwork-11.13educ+2.20age
n=706 and R^2= 0.113
If someone works five more hours per week, by how many minutes sleep is predicted to fall? Is this a large trade off?
Discuss the sign and magnitude of the estimated coefficient on educ
Would you say, totwork, educand age explain much of the variation in sleep? What other factor affect the time spent sleeping? Are these likely to be correlated withtotwork?
To determine the sign of B1, we need to understand the relationship between sleep and total work. In this case, B1 represents the coefficient for total work (totwork) in the regression model. If adults trade off sleep for work, we would expect an inverse relationship between the two variables. In other words, as total work increases, sleep is predicted to decrease. Therefore, the sign of B1 is expected to be negative.
For B2 and B3, which represent the coefficients for education (educ) and age, respectively, it is difficult to determine the signs based solely on the information provided. The signs could be positive, negative, or zero, depending on the specific relationship between these variables and sleep. We would need more contextual information or statistical analysis to make an accurate determination.
To find the predicted decrease in sleep when someone works five more hours per week, we can substitute the given values into the estimated equation. Given the equation (sleep) ̂= 3638.25 - 0.148totwork - 11.13educ + 2.20age, we can calculate the change in sleep by substituting totwork with the new value (totwork + 5) and evaluating the difference in predicted sleep. This will provide the estimate of the fall in sleep minutes.
The magnitude of the trade-off can be assessed by comparing the absolute value of the coefficient for totwork (-0.148) and multiplying it by the change in totwork. If the resulting number is large, it suggests a substantial trade-off, whereas a smaller number indicates a smaller trade-off.
To discuss the sign and magnitude of the estimated coefficient on education (educ), we can observe that the coefficient is negative (-11.13). This suggests that holding other factors constant, an increase in education is associated with a decrease in sleep. However, without further analysis or more contextual information, it is difficult to determine the magnitude of this relationship. It would require evaluating the coefficient in relation to the range of values and the specific unit of measurement for education.
Based on the information provided, the R-squared value of 0.113 implies that totwork, educ, and age together account for approximately 11.3% of the variation in sleep. This means that the model, including these variables, explains only a small portion of the variation in sleep. Therefore, other factors not included in the model likely affect the time spent sleeping. These factors can vary individually and may or may not be correlated with totwork.
To determine if other factors are likely to be correlated with totwork, we would need additional information or conduct further statistical analysis. It is difficult to make conclusive statements on potential correlations without specific knowledge of the data or variables involved.