Flu Vaccination Analysis, Reading the Numbers Honestly
The hardest part of statistical communication is not the calculation. It is the surrounding framing: which question is being answered, what the answer means for a decision-maker, and what the analysis has actually licensed you to conclude. This project ran a series of inference exercises against synthetic clinical data with that framing question always in the foreground.
The problem
Statistical methods often arrive in a portfolio as a set of completed calculations. Confidence intervals are computed. P-values are reported. The reader is left to do the work of understanding what these mean in the context of a decision being made.
The brief here was to run a chain of analyses against synthetic flu-trial data and present each result in language that a non-specialist could act on. The chain covered single-sample proportions, sampling distributions and the Central Limit Theorem, one-tailed and paired t-tests across two sample sizes, and a one-way ANOVA on simulated promotional scenarios. The unifying question across all five was the same: what does this number mean for the next decision someone has to make?
The approach
Five distinct exercises, each anchored to a specific decision context, with the statistical machinery as the vehicle rather than the destination.
Vaccine efficacy and population projection. A binomial inference on placebo-group outcomes (69% flu-free during the trial) and a 95% confidence interval for that proportion (59.94% to 78.06%). Then, using a 40% population vaccination rate and the trial's vaccinated and unvaccinated infection probabilities, projecting that approximately 22% of the wider population would contract flu over the season. Finally, a ten-year binomial calculation showing a 7.02% probability of a consistently vaccinated individual contracting flu three or more times in that decade.
Sample-mean distributions and the CLT. Drew 1,000 observations from an exponential distribution with mean 10. Sample mean and standard deviation aligned with theoretical values (10.22 and 10.49 respectively). Repeated sampling of means with n=2 produced a clearly skewed distribution. The same exercise at n=30 produced an approximately normal distribution. The two histograms side by side are the cleanest visual argument for the CLT a teacher will ever have.
Packaging-design t-test. A one-tailed test on whether a new packaging design (mean attractiveness score 8.2) outperformed the existing design (7.5). Computed t = 1.65 against a critical value of 1.73, with p = 0.0579. The result fell just short of the 5% threshold. The reportable conclusion is not "the new design is no better." It is "the data does not yet support the investment, and a larger sample might or might not change that."
Sample-size sensitivity. Paired t-tests at n=10 and n=30 on bivariate normal data. The smaller sample returned p = 0.0279, the larger p = 0.0019. Both significant, but the larger gave noticeably tighter confidence in the rejection. The exercise illustrates the relationship between sample size and statistical power without overstating it.
Promotional-scenario ANOVA. One-way ANOVA across three promotional strategies. The first run, with simulated SD = 30, returned F = 2.351, p = 0.129. A rerun with SD = 25 returned F = 2.31, p = 0.133. Both tests failed to reject the null hypothesis at the 5% level. The honest readout is that the available data does not show a winner, not that a winner does not exist.
What the exercises taught
Across the five exercises, the recurring lesson was the gap between the calculation and the recommendation. A p-value of 0.0579 is not "almost significant." It is exactly itself, and the decision-maker needs the language to act on it. The 22% projection of population flu rate is contingent on the assumed vaccination uptake, and any communication of it has to carry that assumption with it.
Evidence
- Five statistical methods covered: binomial proportion CI, exponential sampling and the CLT, one-tailed t-test, paired t-test sensitivity, one-way ANOVA
- All exercises reproducible from a single R script with
set.seed(3229642)
- Each result paired with a decision-context interpretation rather than a bare statistical reading
- The packaging-design exercise (p = 0.0579) is the standout case for "the value of being honest about what the test actually says"
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