RES 6003 Module 4 Correlation, Regression and Chi-Square Report Example

How Strongly Do Grades Predict the Exit Examination? Correlation, Simple Linear Regression and Chi-Square Tests of Independence in a Composite Nursing Program

Student Name

American College of Education

RES6003: Applied Statistics

Module 4 Assignment

Instructor Name

November 2, 2026

Introduction

Three earlier reports on the same 186 Kentucky nursing graduates summarized the data, tested the licensure pass rate against a target and compared group means. This report turns to relationships. The faculty want to know how closely nursing GPA and clinical ratings track exit examination scores, whether GPA can usefully predict a student's exit score before the final semester and whether licensure results are related to work hours or to taking the remediation course. Continuous pairs call for correlation and regression; categorical pairs call for chi-square tests of independence.

Correlation Between GPA and Exit Score

Pearson's correlation between nursing GPA and exit examination score was r(184) = .55, p < .001, 95% CI [.44, .64]. An r near .50 is conventionally called a large association (Cohen, 1992). A scatterplot showed a roughly linear, evenly spread cloud of points with no curvature and no single point pulling the line, so Pearson's coefficient was appropriate. Squaring r shows that GPA and exit score share about 30% of their variance, which means that 70% of the variation in exit scores lies in factors GPA does not capture.

Clinical rating correlated more weakly with exit score, r(184) = .22, p = .003, 95% CI [.08, .35], and with GPA, r(184) = .30, p < .001. Schober et al. (2018) cautioned that the descriptive labels attached to correlation sizes are arbitrary and that context should guide interpretation. Here the context matters: clinical ratings bunch near the top of their scale, as Module 1 showed, and a restricted range weakens any correlation, so the low figure may say as much about the rating tool as about the link between clinical and test performance.

Regression of Exit Score on GPA

A simple linear regression with nursing GPA predicting exit score gave the equation: predicted exit score = 556.2 + 121.6 × GPA. The slope was significant, b = 121.6, SE = 13.7, t(184) = 8.88, p < .001, 95% CI [94.6, 148.7], and the model explained 30% of the variance, R² = .30. Each additional tenth of a grade point was associated with about 12 more points on the exit examination. A student with a 2.80 GPA would have a predicted exit score of about 897, and one with a 3.50 GPA about 982.

The standard error of the estimate, 59.3 points, is the figure that keeps this regression honest: a student predicted to score 897 could plausibly score anywhere from about 780 to over 1,000. The model describes the average relationship well, but it is a poor tool for forecasting any one student's score. The residuals showed no pattern against predicted values, which supports the assumptions of linearity and constant variance.

Checking the Regression Assumptions

Simple regression rests on four assumptions, and each was checked against the data. Linearity was supported by the scatterplot and by a residual plot with no curve. Constant variance held reasonably well: the spread of residuals was similar for students with low, middle and high GPAs, with standard deviations of 59.8, 59.2 and 58.0 points across the lowest, middle and highest thirds of GPA and no funnel shape. The residuals were close to normally distributed in a histogram, with a skewness of 0.12, which matters mainly for the slope's confidence interval and test, and with 186 cases small departures would have little effect. Independence is supported by the design, since each graduate contributes one observation, although students taught by the same instructors may share influences the model does not represent.

Using the Regression in Advising

Two cautions apply if faculty want to use the equation. First, it should not be applied outside the range of the data. No graduate in the sample had a GPA below 2.40, because the progression rule removes those students, so predictions for lower GPAs would be extrapolation. Second, the equation was built on graduates only. Students who left the program are missing, and if they differed systematically from graduates the relationship could look different in the full group of entering students. A sensible use is to flag students whose GPA after the second semester predicts an exit score near the 850 benchmark, so that they can be offered support early, without treating the prediction as a verdict.

Chi-Square: Work Hours and Licensure Result

Pass rates were 89.8% (53 of 59) in the lightest work group, 91.4% (64 of 70) in the middle group and 84.2% (48 of 57) among the heaviest workers. A chi-square test of independence found no significant association between work-hour group and licensure result, χ²(2, N = 186) = 1.74, p = .42, Cramér's V = .10. All expected cell counts were above 5, the smallest being 6.4, so the chi-square approximation was appropriate.

This result sits alongside Module 3's finding that heavy workers had lower GPAs. The lower pass rate among heavy workers points in the expected direction, but with only 21 failures spread across three groups, the test had little ability to detect a difference of this size. The absence of a significant result is not evidence that work hours are unrelated to licensure success.

Remediation and Licensure: When Chi-Square Fails

Among all graduates, 28 of the 31 who took the remediation course passed, 90.3%, against 137 of 155 who did not, 88.4%. Only 3.5 failures were expected among remediated students, and with an expected count that low the chi-square approximation cannot be trusted. Kim (2017) explained that Fisher's exact test, which computes the probability directly rather than approximating it, should replace chi-square when expected counts are small. Fisher's exact test found no association, p > .99.

The faculty's sharper question concerns the 15 graduates who scored below the 850 benchmark, the students the remediation course targets. Of the 11 in that group who took remediation, 9 passed; of the 4 who did not, 2 passed. A chi-square statistic for this table would be 1.52, but three of its four expected counts fall below 5, one as low as 1.1, so the figure means little. Fisher's exact test gave p = .52. The data are compatible with remediation helping, hurting or doing nothing, and 15 students cannot settle the question.

What These Results Can Support

Taken together, the tests show one strong, well-estimated relationship, between GPA and exit scores, and several questions the data cannot answer. That is itself useful. It tells the program that GPA can serve as an early signal and that judging the remediation course will require more students, a comparison group chosen more carefully or a different design.

Reporting the Results

In the submitted report, a correlation table gives the coefficients among GPA, clinical rating and exit score with their significance, and a regression table gives the intercept and slope with standard errors, t values, p values and confidence intervals, along with R² and the standard error of the estimate. The chi-square results are reported with degrees of freedom, sample size, the statistic, p and Cramér's V, and the Fisher's exact results with their p values and a note explaining why the exact test replaced the approximation.

Conclusion

Nursing GPA correlated strongly with exit examination scores, r = .55, and predicted them with R² = .30, though with a standard error of about 59 points for individual students. Licensure results were not significantly related to work hours, χ²(2) = 1.74, p = .42, or to remediation, Fisher's exact p = .52 among low scorers, with small samples limiting both tests. Module 5 brings these methods together in an analysis plan for a full study.

References

Cohen, J. (1992). A power primer. Psychological Bulletin, 112(1), 155-159. https://doi.org/10.1037/0033-2909.112.1.155

Kim, H.-Y. (2017). Statistical notes for clinical researchers: Chi-squared test and Fisher's exact test. Restorative Dentistry & Endodontics, 42(2), 152-155. https://doi.org/10.5395/rde.2017.42.2.152

Schober, P., Boer, C., & Schwarte, L. A. (2018). Correlation coefficients: Appropriate use and interpretation. Anesthesia & Analgesia, 126(5), 1763-1768. https://doi.org/10.1213/ANE.0000000000002864