Warm Up

Andrew Dickinson

Sleep and Exam Performance

Consider a dataset obtained from a study on education that investigates the impact of sleep duration on students’ exam performance. The dataset includes a random sample of students in a specific educational institution. The following regression equation estimates the relationship between exam performance (measured by the percentage of maximum possible score) and hours of sleep:

\[ \text{Performance}_i = \beta_0 + \beta_1 \times \text{SleepHours}_i + u_i \]

From the regression output, you have the following estimates:

\[ \text{Performance} = 50 + 5 \times \text{SleepHours} \]

1. Interpret the Estimates: Interpret the intercept and slope coefficients in the context of the model.

1. The intercept coefficient (\(\beta_0 = 50\)) suggests that a student who sleeps zero hours before an exam is expected to score 50% of the maximum possible score on average. The slope coefficient (\(\beta_1 = 5\)) indicates that for each additional hour of sleep, a student’s exam performance is expected to increase by 5% on average.

2. Predicted Outcome: If a student sleeps for 8 hours, what is the predicted exam performance according to the model?

2. Using the model, if a student sleeps for 8 hours, the predicted exam performance is \(50 + 5 \times 8 = 90%\) of the maximum possible score.

3. Effect of Changing X: Suppose a student is considering adjusting their sleeping habits. They currently sleep 6 hours and are thinking of sleeping 8 hours instead. What would the model predict the change in their exam performance to be? In other words, what is the expected increase in exam performance if they sleep for 2 more hours?

3. If a student increases their sleep from 6 to 8 hours, the model predicts an increase in exam performance by \(5 \times 2 = 10%\). This means that by sleeping 2 more hours, their performance could increase from 80% to 90% of the maximum possible score.

4. Two-Sided Hypothesis Test: Test the hypothesis that additional hours of sleep have no effect on exam performance against the alternative that they do have an effect. Use a significance level of 0.05. The critical t-value for with appropriate degrees of freedom is approximately 1.96 for a two-tailed test.

4. For the two-sided hypothesis test \(H_0: \beta_1 = 0\) against \(H_1: \beta_1 \neq 0\), calculate the t-statistic:

\[ t = \frac{\beta_1 - 0}{\text{SE}_{\beta_1}} = \frac{5 - 0}{1.5} \approx 3.33 \]

The critical t-value for \(\alpha = 0.05\) with appropriate degrees of freedom is approximately 1.96 for a two-tailed test. Since the absolute value of the calculated t-statistic exceeds the critical value, we reject the null hypothesis, indicating that additional hours of sleep significantly affect exam performance.

5. One-Sided Hypothesis Test with Minimum Effect Size: Test the hypothesis that additional hours of sleep increase exam performance by no more than 3% per hour against the alternative that each additional hour of sleep increases exam performance by more than 3%. Use a significance level of 0.05. The critical value for a significance level of 0.05 with appropriate degrees of freedom is approximately 1.645.

5. For this one-sided hypothesis test, we use the null hypothesis \(H_0: \beta_1 \leq 3\) against the alternative hypothesis \(H_1: \beta_1 > 3\). The test statistic is calculated as follows:

\[ t = \frac{\beta_1 - 3}{\text{SE}_{\beta_1}} = \frac{5 - 3}{1.5} \approx 1.33 \]

Since the calculated t-statistic is less than the critical value, we fail to reject the null hypothesis, indicating that we do not have sufficient evidence to conclude that each additional hour of sleep increases exam performance by more than 3%.

6. Confidence Intervals: Calculate the 95% confidence interval for \(\beta_1\). Does your confidence interval satisfy the hypothesis test from part d?

6. To calculate the 95% confidence interval for \(\beta_1\), we use the formula:

\[ \text{CI} = \beta_1 \pm z^* \times \text{SE}_{\beta_1} \]

Where \(\beta_1 = 5\), \(\text{SE}_{\beta_1} = 1.5\), and \(z^*\) (the critical value for a 95% confidence level) is approximately 1.96. This results in:

\[ \text{CI} = 5 \pm 1.96 \times 1.5 = (5 \pm 2.94) = (2.06, 7.94) \]

The 95% confidence interval for \(\beta_1\) is (2.06, 7.94). This interval includes values greater than zero but also less than the 3% increase considered in part e. Therefore, this confidence interval supports the conclusion of part d, indicating that there is significant evidence at the 5% level that additional hours of sleep have a positive effect on exam performance, though it does not conclusively support the more specific alternative hypothesis that the effect is greater than 3% per hour.