PSY 315 Week 3 Hypothesis Testing and t Tests Example

Do Short Sleepers Score Lower? Hypothesis Testing and Three t Tests on Class Survey Data

[Student Name]

University of Phoenix

PSY/315: Statistical Reasoning in Psychology

Week 3 Assignment

[Instructor Name]

[Date]

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

Description in Week 1 and probability in Week 2 set up the central task of inferential statistics: deciding whether a pattern in a sample is likely to reflect something real in the population or could easily have arisen by chance. This report applies hypothesis testing to three questions from the composite class survey and reports each result with its effect size and confidence interval.

The Logic of a Significance Test

Every significance test starts from a null hypothesis, typically the claim of zero difference or zero effect, set against an alternative claiming some effect exists. The researcher sets an alpha level, commonly .05, collects data and computes a test statistic. If the result would be very unlikely under the null hypothesis, with p below alpha, the null is rejected. Two mistakes are possible. A Type I error rejects a true null hypothesis, a false alarm. A Type II error fails to reject a false one, a miss. Lowering alpha reduces false alarms but raises the risk of misses unless the sample is larger.

Question One: Sleep and Exam Scores

The survey split students into 42 who slept fewer than six hours and 78 who slept six or more. Under the null, short and longer sleepers share one population mean; under the alternative, their means are not the same. Because the groups contain different students, an independent-samples t test fits.

Short sleepers averaged 72.10 (SD = 11.80), and longer sleepers averaged 78.60 (SD = 10.30). The pooled standard deviation was 10.84, and the standard error of the difference was 2.08. The test gave t(118) = 3.13, p = .002. The difference of 6.50 points had a 95 percent confidence interval from 2.39 to 10.61, and Cohen's d was 0.60, a medium effect.

Interpreted carefully, short sleepers scored lower, and the gap is large enough to matter for a letter grade. The design does not show that short sleep causes lower scores. Students who work night jobs, care for children or feel more stressed might both sleep less and study less.

Checking Assumptions

The independent t test assumes roughly normal scores within each group and similar variances. Histograms of exam scores in both groups were close to bell-shaped, and the two standard deviations, 11.80 and 10.30, differ by much less than a factor of two. With groups of this size, the test is robust to modest departures. Rerunning the comparison with Welch's correction for unequal spread gave nearly the same result.

Question Two: The Class Against a Benchmark

The department reports that students across the college average 74 on this common exam. A one-sample t test asks whether this class differs. With a mean of 76.30, a standard deviation of 11.20 and 120 students, the standard error is 1.02, and t(119) = 2.25, p = .026. The effect size, d = 0.21, is small. With a large sample, even a modest difference reaches significance, which is why the effect size matters: the class did slightly better than the college as a whole, not dramatically better.

A p value of .026 says the class probably differs from the benchmark; an effect size of 0.21 says the difference is modest.

Question Three: Stress Before and After a Breathing Exercise

In one section, 25 students rated their stress before and after a five-minute guided breathing exercise at the start of class. Because each student gave two ratings, a paired t test compares the differences within each person. Ratings fell from an average of 6.40 to 5.60, a mean change of 0.80 with a standard deviation of 1.50. The test gave t(24) = 2.67, p = .013, with an effect size of 0.53 based on the differences.

Two cautions apply. Stress ratings are ordinal, so a Wilcoxon signed-rank test would be a reasonable check; it gave the same conclusion. And without a comparison group that sat quietly for five minutes, the drop might reflect settling into class rather than the breathing itself.

What p Values Can and Cannot Say

Cohen (1994) criticized the habit of treating p below .05 as the goal of research, pointing out that a small p does not tell a researcher how likely the null is to be correct and that rejecting a null of exactly zero says almost nothing about whether an effect is big enough to care about. He urged researchers to report effect sizes and confidence intervals. Wasserstein and Lazar (2016), writing for the American Statistical Association, set out six principles, among them that p by itself cannot convey effect size or practical weight and that crossing one cutoff such as .05 is a poor basis for a scientific or policy decision.

Lakens (2013) offered a practical guide to computing and reporting effect sizes for t tests and analysis of variance, noting that effect sizes allow results to be compared across studies and combined in meta-analyses. The three tests above follow that advice.

Errors in This Context

A Type I error in question one would mean concluding that short sleepers score lower when, in the population, they do not; the college might launch a sleep campaign aimed at a problem that does not exist. A Type II error in question three would mean missing a real benefit of the breathing exercise because 25 students were too few to detect it, and instructors might abandon a helpful routine.

Writing the Results Section

A results paragraph for question one might read: students who slept fewer than six hours scored lower on the exam (M = 72.10, SD = 11.80) than students who slept six hours or more (M = 78.60, SD = 10.30), t(118) = 3.13, p = .002, d = 0.60, 95% CI [2.39, 10.61]. Each statistic is italicized in the final paper, means and standard deviations come first and the sentence states the direction of the difference before any number, so a reader learns the finding even if they skip the statistics.

Conclusion

Three t tests matched to three designs show that short sleepers in this survey scored about six and a half points lower, that the class modestly outperformed the college benchmark and that stress ratings dropped after a short breathing exercise. Reporting effect sizes and confidence intervals with each p value, checking assumptions and resisting causal claims make these findings informative rather than merely significant.

References

Cohen, J. (1994). The earth is round (p < .05). American Psychologist, 49(12), 997-1003. https://doi.org/10.1037/0003-066X.49.12.997

Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science: A practical primer for t-tests and ANOVAs. Frontiers in Psychology, 4, Article 863. https://doi.org/10.3389/fpsyg.2013.00863

Wasserstein, R. L., & Lazar, N. A. (2016). The ASA statement on p-values: Context, process, and purpose. The American Statistician, 70(2), 129-133. https://doi.org/10.1080/00031305.2016.1154108