Sleep and GPA Survey
A student council surveys 240 students during lunch about sleep and grades. Of 90 students who reported sleeping at least 7 hours per night, 63 reported a GPA of 3.0 or higher. Of the 150 students who reported sleeping fewer than 7 hours, 45 reported a GPA of 3.0 or higher. The council plans to publish the headline “More sleep raises GPA.” A statistics teacher notes that the survey was a volunteer lunch-time sample, that GPA was self-reported, and that students who sleep less may also work more hours. The council must decide whether to run the headline, what conditional probabilities to report, and whether the data support a causal claim.
1. Build the two-way table and compute the conditional probability of GPA ≥ 3.0 for each sleep group. Also compute the relative risk and interpret it.
Read the worked answer
The two-way table is:
| GPA ≥ 3.0 | GPA < 3.0 | Total | |
|---|---|---|---|
| Sleep ≥ 7 h | 63 | 27 | 90 |
| Sleep < 7 h | 45 | 105 | 150 |
| Total | 108 | 132 | 240 |
Conditional probabilities:
$$P(\text{GPA}\ge3.0\mid\text{sleep}\ge7)=\frac{63}{90}=0.70,$$
$$P(\text{GPA}\ge3.0\mid\text{sleep}<7)=\frac{45}{150}=0.30.$$
The relative risk is $0.70/0.30\approx2.33$. Students in the high-sleep group were about 2.3 times as likely to report a GPA of at least 3.0. The overall rate is $108/240=0.45$, which is between the two conditional rates, as expected.
2. Does the study support the headline “More sleep raises GPA”? Identify confounders and explain what design change would be needed for a causal claim.
Read the worked answer
No. The data show a strong association, but association is not causation. Several confounders could produce both more sleep and higher GPA: students who work fewer hours have more time to sleep and study; students with better health or less anxiety may sleep more and perform better; households with more resources may provide quieter bedrooms and tutoring; and students who use less late-night screen time may sleep more and study more effectively. A randomized experiment would be needed to support “raises”: assign students to different sleep-duration conditions while holding study time and other factors comparable, then measure GPA. That design is difficult and ethically sensitive, so the council should use cautious language such as “students who reported more sleep also reported higher GPAs.”
3. Evaluate the sampling method and self-reported GPA. How do these limitations affect the strength of the council’s conclusion?
Read the worked answer
The sample is a volunteer lunch-time sample, not a random sample of the school. Students who eat in the cafeteria, have free lunch periods, or are willing to answer may differ systematically from those who are absent, working, or uninterested. Self-reported GPA may be inflated or misremembered, and the sleep question asks about a typical night without verifying it. These problems create selection bias and measurement error, so the 0.70 and 0.30 rates should not be treated as precise estimates for the whole school. The association is strong enough to justify further investigation, but the council should report the sampling method and avoid publishing a causal headline. A better design would randomly select students, verify GPA from school records with consent, and use a validated sleep questionnaire.