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.
1: If adults trade off sleep for work what is the sign of B_1?
2: What signs do you think of B_2and B_3?
3: Suppose the estimated equation is;
(sleep) ̂=3638.25-0.148totwork-11.13educ+2.20age
n=706 and R^2= 0.113
3: If someone works five more hours per week, by how many minutes sleep is predicted to fall? Is this a large trade off?
4: Discuss the sign and magnitude of the estimated coefficient on educ
5: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 answer the questions, let's analyze the provided multiple regression model and the estimated equation.
1. Sign of B_1:
In the given regression model, the coefficient B_1 represents the relationship between sleep and total work. If adults trade off sleep for work, we can expect a negative relationship between the two variables. Therefore, the sign of B_1 is likely to be negative.
2. Signs of B_2 and B_3:
In the regression model, B_2 represents the relationship between sleep and education, while B_3 represents the relationship between sleep and age. Without additional information, it is difficult to determine the signs of these coefficients. However, one could make an educated guess based on common expectations. For example, higher education levels might be associated with better time management skills, potentially leading to more efficient sleep patterns. This could suggest a positive relationship between sleep and education (B_2 > 0). Regarding age, it is difficult to predict the sign as it can vary depending on individual circumstances.
3. Predicted sleep given the estimated equation:
Based on the provided estimated equation:
(sleep) ̂ = 3638.25 - 0.148totwork - 11.13educ + 2.20age
To calculate the predicted change in sleep when someone works five more hours per week, we substitute the new value into the equation:
predicted sleep change = -0.148 * 5 = -0.74 minutes
Therefore, increase in work by five more hours per week is predicted to result in a decrease in sleep by approximately 0.74 minutes. Whether this trade-off is deemed large or not would depend on individual perspectives and the context of the study.
4. Sign and magnitude of the estimated coefficient on educ:
The estimated coefficient, -11.13, represents the relationship between sleep and education after controlling for other variables. Based on the magnitude and negative sign, it suggests that an increase in education is associated with a decrease in sleep duration. However, the impact of education on sleep should be interpreted cautiously, as the coefficient can be influenced by various factors and does not necessarily imply causation.
5. Explanation of variation in sleep and other factors affecting sleep:
To determine whether totwork, educ, and age explain much of the variation in sleep, we can refer to the coefficient of determination (R^2). In this case, R^2 is given as 0.113, suggesting that the independent variables in the model together explain 11.3% of the observed variation in sleep. While this indicates that the model does capture some of the factors influencing sleep, there is still substantial unexplained variation.
Other factors affecting sleep not included in the model can vary among individuals and may include psychological factors (e.g., stress, anxiety), lifestyle choices (e.g., exercise, diet), health conditions, and environmental factors (e.g., noise, light). Whether these factors are correlated with totwork would depend on the specific circumstances and the underlying relationship between the variables. It is possible that some factors affecting sleep are correlated with total work, while others may be independent.