MPH 550 Week 3 Confidence Intervals and Hypothesis Testing Example

Did Diabetes Really Rise, and Is the East Side Different? Confidence Intervals and Hypothesis Tests With Colorado Survey Data

[Student Name]

University of Phoenix

MPH/550: Public Health Statistics

Week 3 Assignment

[Instructor Name]

[Date]

The county survey, its respondents and results are composites written for a model paper; statewide estimates come from CDC's BRFSS prevalence data, and other findings come from the sources cited.

When the county's household survey results arrived, a health board member asked the question every analyst hears: are these differences real, or are we reading noise? The analyst answered with confidence intervals and hypothesis tests on three questions. This paper presents each test, its interpretation and what it means for the board.

The Logic of Inference

A sample estimate differs from the true population value by chance. A confidence interval gives a range of values compatible with the data under the model. A hypothesis test asks whether the data are surprising if a null hypothesis, usually no difference, were true, measured by the p-value.

Question One: Did Diabetes Rise Statewide?

CDC's BRFSS prevalence data report that 7.0% of Colorado adults had diagnosed diabetes in 2019, with a 95% confidence interval of 6.4% to 7.6%, and 8.76% in 2025, with an interval of 8.13% to 9.39% (Centers for Disease Control and Prevention [CDC], 2026). The null hypothesis states that prevalence was the same in both years; the alternative states that it differed.

The Test

Standard errors can be recovered from the published intervals by dividing their width by 3.92: about 0.31 points for 2019 and 0.32 for 2025. Because the two years drew separate samples, their variances add: squaring each standard error, summing and taking the root yields about 0.44 points for the gap between years. The difference of 1.76 points divided by 0.44 gives z of about 3.97, with a two-sided p-value below 0.001.

Interpreting Question One

The 95% interval for the change runs from 0.89 to 2.63 percentage points. The data are hard to reconcile with no change, and the plausible increase ranges from under one point to over two and a half. The estimates are crude, not age-adjusted, so an aging population could explain part of the rise; the test shows that prevalence changed, not why.

Question Two: Is the County Higher Than the State?

The county survey completed 702 interviews, and 91 respondents reported diagnosed diabetes, 12.96%. The simple standard error is about 1.27 points, but the survey used stratification and weighting, so the analyst applied a design effect of 1.5, multiplying the standard error by its square root to reach about 1.55 points. The 95% interval is 9.9% to 16.0%.

The Comparison

Compared with the state's 8.76% for 2025, the difference is 4.2 points. Combining the county's standard error with the state's gives about 1.59 points, and z is about 2.65, with a p-value of about 0.008. The county interval lies entirely above the state estimate.

Interpreting Question Two

The county's diagnosed diabetes prevalence is very likely higher than the state's, by somewhere between about 1 and 7 points. The survey's age structure matters here too: the county is older than the state, so part of the gap reflects age.

Question Three: Is the East Side Different?

The survey oversampled the east side. There, 47 of 310 respondents reported diabetes, 15.2%, compared with 44 of 392 elsewhere, 11.2%. The difference is 3.9 points. After inflating for the design effect, the uncertainty in that gap is about 3.2 points, z is 1.24 and the p-value is 0.21. The 95% interval runs from about minus 2.3 to plus 10.1 points.

Interpreting Question Three

A board member concluded that the east side is no different. That conclusion does not follow. The interval includes no difference but also differences as large as 10 points. Two statisticians made the point in a short, much-cited note: failing to find evidence of an effect is different from finding evidence that there is none, and a non-significant result from a small study may reflect insufficient data rather than no effect (Altman & Bland, 1995). The survey could not tell whether the east side is the same or much worse.

Type I and Type II Errors

A type I error is a false alarm, rejecting a true null. With a 0.05 threshold, a true null will be rejected 5% of the time. A type II error misses a real difference. The east side comparison had low power: with about 300 and 400 respondents and the design effect, a real 4-point difference would be detected only about a quarter of the time.

Power

Power depends on the size of the true difference, the sample size and variability. To detect a 4-point difference with 80% power, the survey would need several times as many east side respondents. The analyst recommended pooling two survey years before drawing conclusions about the east side.

What P-Values Do Not Say

Misreadings are common. The American Statistical Association issued six principles on the subject. In summary, a p-value signals how poorly the data fit a stated model; it is not the chance that the null is correct, nor the chance that luck alone produced the data. Decisions ought not to hinge on crossing a cutoff, and a small p-value says nothing about how large or how meaningful an effect is (Wasserstein & Lazar, 2016). Full reporting and transparency, the statement adds, are needed for proper inference.

More Misinterpretations

A guide by epidemiologists and statisticians cataloged common errors, including treating a non-significant result as evidence for the null, treating a 95% interval as having a 95% chance of containing the true value in a specific study and comparing studies by whether each crossed the significance line (Greenland et al., 2016). The analyst avoided these in her report.

Why Intervals Come First

The analyst led with intervals because they carry more information than a p-value. The statewide interval shows both that the rise is unlikely to be zero and that it is probably small. The east side interval shows that the question is open. A p-value alone would have reduced each result to a yes or no, hiding exactly the information the board needed.

Multiple Questions

The analyst ran three planned tests, each answering a distinct question agreed before the data arrived. She did not search the survey for every possible subgroup difference, which would have produced false alarms by chance. Any additional comparisons will be labeled exploratory.

Reporting to the Board

The report gave each estimate with its interval first and p-values second. Statewide diabetes rose modestly; the county is higher than the state; the east side may be higher, but the survey is too small to say. Each conclusion carried a plain-language sentence about confidence.

Assumptions

The tests assume independent samples, approximately normal sampling distributions, which holds with these sample sizes, and correct design effects. The design effect of 1.5 is an estimate; the final weighted analysis will calculate it directly.

Limits

Self-reported diagnosed diabetes misses undiagnosed cases. The comparisons are crude, not age-adjusted. The state and county surveys used different methods, which could produce differences unrelated to true prevalence.

Conclusion

Confidence intervals and hypothesis tests answered the board member's question in three ways. The statewide rise is real but modest, the county is likely higher than the state and the east side comparison is inconclusive rather than negative. Reporting intervals with p-values, and heeding statisticians' guidance, kept the analysis from claiming more or less than the data showed.

References

Altman, D. G., & Bland, J. M. (1995). Absence of evidence is not evidence of absence. BMJ, 311(7003), 485. https://doi.org/10.1136/bmj.311.7003.485

Centers for Disease Control and Prevention. (2026). Behavioral Risk Factor Surveillance System (BRFSS) prevalence data (2011 to present) [Data set]. https://data.cdc.gov/d/dttw-5yxu

Greenland, S., Senn, S. J., Rothman, K. J., Carlin, J. B., Poole, C., Goodman, S. N., & Altman, D. G. (2016). Statistical tests, P values, confidence intervals, and power: A guide to misinterpretations. European Journal of Epidemiology, 31(4), 337-350. https://doi.org/10.1007/s10654-016-0149-3

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