Skewed Minutes and a Fair Test: Choosing Between the t-Test, the Mann-Whitney Test and the Chi-Square Test for Door-to-Provider Times Under Two Triage Models
Student Name
American College of Education
RES5303: Research Methods and Applied Statistics in Healthcare
Module 5 Assignment
Instructor Name
October 4, 2027
The Question and the Data
A community emergency department, a composite invented for this assignment, tested a provider-in-triage model for three months, in which a physician assistant sees lower-acuity patients at triage, and compared it with its standard model, in which patients wait for a treatment space before seeing a provider. The operations manager wants to know whether door-to-provider time differed between the two models. The data are illustrative but shaped like real emergency department data: 1,198 visits under the provider-in-triage model and 1,240 visits under the standard model, with the times recorded from arrival to first provider contact in the electronic record.
The data are strongly right-skewed, as waiting times almost always are. Under the standard model, the median was 38 minutes, the mean 52 minutes and the standard deviation 41 minutes, with a long tail of visits over two hours on busy evenings. Under the provider-in-triage model, the median was 24 minutes, the mean 33 minutes and the standard deviation 29 minutes. A histogram of each group shows most visits bunched at the low end and a tail stretching to the right, and the mean sits well above the median in both groups. Before choosing a test, a researcher has to look at the data, because every test makes promises about the data that the data may not keep.
Candidate One: The Independent-Samples t-Test
The independent-samples t-test compares the means of two independent groups. Its assumptions are that observations are independent, that the outcome is measured on an interval or ratio scale, that the sampling distribution of the difference in means is approximately normal and, for the classic Student version, that the two groups have equal variances. Door-to-provider time is a ratio variable, and visits are largely independent, though a small number of patients visited more than once, a point to address by keeping only each patient's first visit in the period.
The normality assumption is the one that skewed data seem to violate. What the test requires, however, is approximate normality of the sampling distribution of the mean, not of the raw data, and with more than a thousand observations per group the central limit theorem makes that assumption reasonable even for skewed times. Fagerland (2012) examined this question directly and concluded that for studies with large samples, t-tests and their confidence intervals can and should be used even for heavily skewed data. The equal variance assumption is not met, since the standard deviations are 41 and 29 minutes, so the right form here is the Welch adjustment, which drops the equal-variance requirement; Delacre et al. (2017) argue that Welch's test should be the default for this reason. The t-test answers a clear operational question: how many minutes, on average, did the model change the wait? That is also the figure a staffing planner can use directly, since average minutes saved multiplied by visits gives the hours of waiting removed from the department each month.
Candidate Two: The Mann-Whitney Test
The Mann-Whitney test, also called the Wilcoxon rank-sum test, is often described as the nonparametric alternative to the t-test for skewed data. Its assumptions are independent observations and an outcome that can be ranked. It makes no assumption about normality, which is why it is frequently chosen for waiting times.
The difficulty is that the Mann-Whitney test does not compare means, and it only compares medians under the additional assumption that the two distributions have the same shape. Here they do not; the standard model's distribution is more spread out. Fagerland (2012) showed that in large samples with skewed data and different spreads, the Mann-Whitney test can produce very small p values that reflect the probability that a randomly chosen visit from one group is shorter than a randomly chosen visit from the other, rather than a difference in average time. That is a legitimate question, but it is not the operations manager's question. Choosing a test is not only about which assumptions hold; it is about which question the test answers.
Candidate Three: The Chi-Square Test for a Target
Emergency departments often care less about the average wait than about the proportion of patients seen within a target time. If the department's target is to see patients within 30 minutes of arrival, the question becomes whether the proportion meeting the target differs between the two models. In the illustrative data, 41 percent of standard-model visits and 62 percent of provider-in-triage visits met the 30-minute target.
The chi-square test of independence compares proportions across groups. Its assumptions are that observations are independent, that each observation falls into exactly one category and that the expected count in each cell is large enough, commonly at least five, with Fisher's exact test used when expected counts are small (Kim, 2017). With more than a thousand visits in each group, every expected count is in the hundreds, so the assumption is easily met. The chi-square test, reported with the difference in proportions and its confidence interval, answers the target question directly and in terms a department leader already uses.
The Recommended Analysis
The recommended plan reports two procedures, each answering a different question, both chosen before the results were examined. The primary analysis is Welch's t-test on door-to-provider time, restricted to each patient's first visit, reported as the difference in mean minutes with its 95 percent confidence interval; with these data, the difference is about 19 minutes. The secondary analysis is a chi-square test of the proportion seen within 30 minutes, reported as a difference of about 21 percentage points with its confidence interval. Medians and interquartile ranges will be reported descriptively for both groups, because they describe the typical patient's experience better than means in skewed data.
The Mann-Whitney test will not be used as the main test, because its question, the probability that one visit is shorter than another, is not the question the department asked, and its result in a sample this large could be misread as evidence about medians. Polit and Beck (2021) emphasize that statistical procedures should be selected based on the research question, the level of measurement and the assumptions, and reporting the reasoning for each choice lets readers judge the analysis rather than trust it.
Conclusion
Door-to-provider times are skewed, and skewness alone does not decide the test. With large samples, Welch's t-test gives a valid comparison of mean waiting time despite the skew and the unequal variances; the Mann-Whitney test answers a different question than the one the department asked; and the chi-square test answers the target-based question that emergency leaders care about. Checking each assumption against the data and matching each procedure to its question produces an analysis that is both correct and useful: the provider-in-triage model reduced the average wait by about 19 minutes and increased the share of patients seen within 30 minutes by about 21 percentage points.
References
Delacre, M., Lakens, D., & Leys, C. (2017). Why psychologists should by default use Welch's t-test instead of Student's t-test. International Review of Social Psychology, 30(1), 92-101. https://doi.org/10.5334/irsp.82
Fagerland, M. W. (2012). t-tests, non-parametric tests, and large studies: A paradox of statistical practice? BMC Medical Research Methodology, 12, Article 78. https://doi.org/10.1186/1471-2288-12-78
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
Polit, D. F., & Beck, C. T. (2021). Nursing research: Generating and assessing evidence for nursing practice (11th ed.). Wolters Kluwer.
How this RES 5303 Module 5 example is structured
RES 5303 Module 5 often works specific procedures and the assumptions deciding which one applies; your classroom's instructions decide the procedures and whether real or illustrative data are used. This example describes the question and the data first, including their shape, because assumptions can only be checked against actual data. It then takes each candidate procedure in turn, states its assumptions, checks them and explains what the procedure actually tests. The final section recommends an analysis plan and explains why two procedures answering different questions are reported together rather than one chosen to give the smallest p value.
RES5303 Module 5 questions, answered
What does RES5303 Module 5 usually ask for?
RES5303 Module 5 often asks students to select statistical procedures for a healthcare question and justify the choice by checking assumptions such as level of measurement, independence, normality and equal variances. Some sections provide a dataset; others ask for an analysis plan. Your classroom's instructions decide which procedures to cover and whether calculations are required.
Can I use a t-test when my data are skewed?
Often yes, when the sample is large, because the t-test depends on the sampling distribution of the mean being approximately normal, which the central limit theorem makes likely in large samples. Use Welch's version if the group variances differ. For small skewed samples, a nonparametric test or a transformation may be more appropriate.
Does the Mann-Whitney test compare medians?
Only if the two distributions have the same shape. Otherwise it tests whether a randomly chosen value from one group tends to be larger than one from the other. That can be a useful question, but it is different from comparing averages or medians, so choose it only when it matches what you want to know.
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