PSY 315 Week 5 Correlation, Regression and Chi-Square Example

Sleep, Stress and Scores: Correlation, Regression and a Chi-Square Test on One Class Survey

[Student Name]

University of Phoenix

PSY/315: Statistical Reasoning in Psychology

Week 5 Assignment

[Instructor Name]

[Date]

The survey, the students and all numbers are composites written for a model paper; statistical guidance and research findings come from the sources listed.

Earlier weeks described the class survey, located scores on the normal curve and compared group means. The final step asks how variables move together. Correlation measures the strength of a relationship, regression turns it into a prediction and chi-square tests relationships between categories. This report applies all three to the composite survey and closes by connecting the course's tools.

Look Before Computing

Anscombe (1973) built a quartet of eleven-point data sets that share almost the same summary numbers and fitted line, yet their scatterplots looked completely different: one linear, one curved, one with a single outlier and one driven entirely by a single extreme point. His point, that numbers should never replace a graph, is the reason this analysis begins with a scatterplot of sleep hours against exam scores. The plot shows a loose upward drift, no obvious curve and no single point controlling the pattern, so a Pearson correlation is reasonable.

Sleep and Exam Scores: Pearson's r

The correlation between hours of sleep and exam scores was r(118) = .32, p < .001. The relationship is positive and moderate: students who slept more tended to score higher, though with much scatter. Squaring r gives .10, meaning sleep accounts for about 10 percent of the variability in exam scores and 90 percent depends on other things, such as study time, prior knowledge and test anxiety.

Predicting Scores From Sleep

A simple regression fits the line that best predicts exam scores from sleep. The slope equals r multiplied by the ratio of the standard deviations, .32 times 11.2 divided by 1.1, or 3.26. The intercept is 76.3 minus 3.26 times 6.4, or 55.44. The equation predicts an exam score of 55.44 plus 3.26 points per hour of sleep.

A student who sleeps five hours would be predicted to score about 71.7, and one who sleeps eight hours about 81.5. The standard error of estimate, about 10.7 points, shows how far actual scores typically fall from these predictions, so individual forecasts are rough. Predictions should stay within the observed range of four to nine hours; the line says nothing reliable about students who sleep two hours or twelve.

What the Research Says

Pilcher and Huffcutt (1996) combined experimental studies of sleep deprivation and found that losing sleep substantially impaired cognitive performance, motor performance and mood, with mood most affected. Because those studies manipulated sleep, they offer causal evidence that the survey cannot. Hershner and Chervin (2014) reviewed research on college students and reported that daytime sleepiness is widespread, driven by irregular schedules, technology use, work and caffeine, and is linked to lower grades and impaired learning. Together these sources make it plausible that sleep contributes to the pattern in the survey, while the survey's own correlation remains open to other explanations.

A correlation of .32 means sleep matters to exam scores, and it also means ninety percent of the story lies somewhere else.

Caffeine and Stress: Spearman's Rho

Caffeine is strongly skewed and stress is ordinal, so Pearson's r would be distorted by the few heavy drinkers. Spearman's rho correlates the ranks instead. The result was a small positive relationship, rho(118) = .21, p = .021: students who drank more caffeine tended to report somewhat more stress. The direction is unclear; stressed students may reach for coffee as easily as coffee may raise stress.

Short Sleep and High Stress: Chi-Square

To test whether two categories are related, the survey's counts form a two-by-two table. Of 42 short sleepers, 26 reported high stress and 16 did not; of 78 longer sleepers, 19 reported high stress and 59 did not. If sleep and stress were independent, expected counts would be 15.75 short sleepers with high stress, 26.25 without, 29.25 longer sleepers with high stress and 48.75 without. Every expected count exceeds five, so the test is appropriate.

Summing the squared differences between observed and expected counts, each divided by the expected count, gives chi-square(1, N = 120) = 16.42, p < .001. The phi coefficient, .37, indicates a moderate association. High stress was far more common among short sleepers than independence would predict.

Limits of the Analysis

All three relationships come from one survey of students at one college, and all rely on self-reports. Students who sleep less may also work more hours, care for family or live farther from campus, and any of these could affect both sleep and grades. The findings describe associations worth testing with stronger designs, such as the experiments Pilcher and Huffcutt reviewed.

Reporting the Results Together

A combined results paragraph might read: sleep correlated positively with exam scores, r(118) = .32, p < .001, and each additional hour predicted about 3.3 more points; caffeine correlated weakly with stress, rho(118) = .21, p = .021; and short sleep was associated with high stress, chi-square(1, N = 120) = 16.42, p < .001, phi = .37. Ordering the results from strongest to weakest evidence, and naming each test before its numbers, helps readers follow three analyses in one paragraph without losing track of which variables each one involved.

Connecting the Course

The same survey has now been described, placed on the normal curve, tested for group differences and examined for relationships. Each tool asked a different question of the same 120 students, and each required matching the method to the variable's scale and shape. Descriptive statistics showed caffeine's skew, which in turn called for medians in Week 1 and Spearman's rho here. The standard error from Week 2 sits inside every t, F and r test, and the effect sizes reported since Week 3 kept attention on how large each pattern was, not only on whether it cleared a threshold of significance.

Conclusion

Sleep and exam scores showed a moderate positive correlation that supports a modest prediction, caffeine and stress a small rank-based relationship and short sleep and high stress a clear association in counts. Plotting first, choosing statistics that fit the data, reporting effect sizes and avoiding causal language turn these numbers into honest findings.

References

Anscombe, F. J. (1973). Graphs in statistical analysis. The American Statistician, 27(1), 17-21. https://doi.org/10.1080/00031305.1973.10478966

Hershner, S. D., & Chervin, R. D. (2014). Causes and consequences of sleepiness among college students. Nature and Science of Sleep, 6, 73-84. https://doi.org/10.2147/NSS.S62907

Pilcher, J. J., & Huffcutt, A. I. (1996). Effects of sleep deprivation on performance: A meta-analysis. Sleep, 19(4), 318-326. https://doi.org/10.1093/sleep/19.4.318