Question

Using the data in SLEEP75 (see also Problem 3 in Chapter 3), we obtain the estimated equation $$\begin{aligned}\widehat{\text {sleep}}=& 3,840.83-.163 \text {totwrk}-11.71 \text {educ}-8.70 \text { age} \\\&(235.11)\quad(.018) \quad (5.86) \quad (11.21)\\\&+.128 \text { age}^{2}+87.75 \text { male } \\ &(.134) \quad (34.33)\\\n=& 706, R^{2}=.123, \bar{R}^{2}=.117\end{aligned}$$ The variable sleep is total minutes per week spent sleeping at night, totwr \(k\) is total weekly minutes spent working, educ and age are measured in years, and male is a gender dummy. i. All other factors being equal, is there evidence that men sleep more than women? How strong is the evidence? ii. Is there a statistically significant tradeoff between working and sleeping? What is the estimated tradeoff? iii. What other regression do you need to run to test the null hypothesis that, holding other factors fixed, age has no effect on sleeping?

Short answer

i. Yes, there is strong evidence that men sleep more than women. ii. Yes, there is a significant tradeoff of -0.163 minutes less sleep per work minute. iii. Test age and age squared effects jointly using a null hypothesis test.

Step-by-step solution

Analyze the Coefficient of Male

To determine if men sleep more than women, we examine the coefficient of the "male" dummy variable, which is 87.75. This implies that, holding all other factors constant, men sleep 87.75 minutes more per week than women.

Evaluate Statistical Significance of Gender Effect

To assess the strength of evidence, compare the estimate 87.75 against its standard error, 34.33, using a t-test. The t-statistic is calculated as \( \frac{87.75}{34.33} \approx 2.56 \). A t-statistic with an absolute value greater than 2 is typically considered statistically significant, suggesting evidence that men sleep more.

Analyze the Coefficient of Total Work

The coefficient for "totwrk" is -0.163, indicating that for each additional minute spent working per week, sleep decreases by 0.163 minutes, assuming other variables remain constant.

Evaluate Statistical Significance of Work-Sleep Tradeoff

To determine the significance of the tradeoff, compute the t-statistic for "totwrk" as \( \frac{-0.163}{0.018} \approx -9.06 \). Since this is much greater in magnitude than 2, it indicates a statistically significant negative relationship between work and sleep.

Test for Age Effect on Sleep

To test if age affects sleep, the null hypothesis is that the coefficients for "age" (\(-8.70\)) and "age^2" (\(+0.128\)) are both zero. Examine their significance by considering their coefficients and standard errors (11.21 for "age" and 0.134 for "age^2"). Additional regression controlling for possible multicollinearity and interaction terms might be needed to confirm findings.

Key concepts

Regression Analysis

Regression analysis is a statistical method used to study the relationships between one dependent variable and one or more independent variables. In the context of the SLEEP75 dataset, the dependent variable is the total minutes of sleep per week.

Independent variables like total work minutes (totwrk), educational attainment (educ), age, and gender (male, as a dummy variable) attempt to explain variation in sleep. Through regression, we can quantify how changes in these predictors affect sleep duration.

• Interpretation: Each coefficient in the regression equation quantifies the change in the dependent variable associated with a one-unit change in the predictor, holding other factors constant.
• Intercept: Represents the expected value of sleep when all predictors are zero.
• Multiple Linear Regression: Allows assessment of complex relationships by including several predictors concurrently.

Regression analysis provides insights into significant predictors influencing sleep, enabling researchers to understand their effects with statistical backing.

Dummy Variables

Dummy variables are binary variables used in regression analysis to represent categorical data numerically. In the exercise, the "male" variable is a dummy variable, coded as 1 for males and 0 for females.

This inclusion helps assess whether gender affects sleep duration. By interpreting the coefficient of the dummy variable, we can determine if there's a significant difference in the dependent variable between the categories represented. For instance,

• The coefficient of 87.75 suggests that, on average, males sleep approximately 87.75 minutes more per week than females, given other variables are held constant.
• This effect is statistically significant if the calculated t-statistic is relatively high, confirming a real difference attributable to gender.

Understanding dummy variables is essential for analyzing and interpreting categorical effects within regression models.

T-Statistic

A t-statistic is a tool used in statistics to test hypotheses about regression coefficients. It compares the observed effect size to the variation in the data, measured by its standard error.

In the SLEEP75 study,

• The t-statistic is calculated using the formula: \[ t = \frac{\text{coefficient}}{\text{standard error}} \]
• For example, the t-statistic for the male coefficient of 87.75 is approximately 2.56, indicating statistical significance (since it's greater than 2).
• The totwrk variable yields a t-statistic of -9.06, showing a significant negative impact on sleep as it deviates notably from zero.

A t-statistic helps determine if a variable’s influence is strong enough to be unlikely due to random chance, crucial for validating model results.

Statistical Significance

Statistical significance indicates the likelihood that a relationship observed in a sample exists in the broader population. It is usually evaluated by a p-value or, indirectly, a t-statistic.

• A variable is statistically significant if its p-value is less than 0.05, implying less than a 5% chance the observed effect is due to random variation.
• The t-statistics for both male and totwrk are greater than the critical value (often around 2), implying their effects on sleep are statistically significant.
• Significance suggests practical importance of predictors, demonstrating their roles beyond coincidence.

Understanding statistical significance is crucial for distinguishing between predictors that truly influence the dependent variable and those that appear to do so by random chance.