PU525 Unit 5 Survey and Rate Confidence Intervals Example

Narrow Enough to Compare, Too Wide to Trust: A Confidence Interval Exercise on Two Estimates From a Composite Rural County, Adult Smoking From a Survey of 1,150 Residents and an Overdose Death Rate Built on Fourteen Deaths, and Why Agencies Flag the Second

[Student Name]

Purdue University Global

PU525: Biostatistics for Public Health

Unit 5 Confidence Interval Exercise

[Instructor Name]

[Date]

The county and every figure are composite and bracketed for the writer's own data; the methods cited are real.

Part One: Adult Smoking Prevalence

Data: [193] of [1,150] respondents reported current smoking. Estimate: [193] / [1,150] = [16.8] percent. Standard error: the square root of p times one minus p over n, about [1.10] percentage points. Critical value for 95 percent: [1.96]. Margin of error: [1.96] times [1.10], about [2.2] points. Interval: [14.6] to [19.0] percent. Comparison: the state's prevalence of [13] percent lies below the whole interval, so the county's higher rate is unlikely to reflect sampling error alone. Meaning: the county can justify concentrating tobacco cessation resources locally, at least among adults like those surveyed.

Part Two: The Overdose Death Rate

Data: [14] overdose deaths in a population of [24,600] in one year. Rate: [14] / [24,600] times [100,000] = [56.9] per [100,000]. Interval: for small counts, an interval based on the Poisson distribution is appropriate rather than the normal approximation (Ulm, 1990). The 95 percent interval for the count runs from about [7.6] to [23.5] deaths, giving a rate interval of about [31.1] to [95.5] per [100,000]. Comparison: the state rate of [32.0] lies inside the interval, so the data cannot show that the county's rate differs from the state's. The county's rate looks nearly twice the state's, yet the interval is wide enough to include the state's rate.

Why Agencies Flag Such Rates

The relative standard error of a count of [14] is about one divided by the square root of [14], or [27] percent. For this reason, national vital statistics reports flag any rate resting on under [20] deaths as unreliable (Xu et al., 2021). A rate resting on so few events can jump or drop sharply between years for no reason other than chance, and publishing it without a warning invites false alarms or false reassurance.

Pooling Years

Five years combined: [58] deaths over [123,000] person-years, a rate of [47.2] per [100,000] with an interval of about [35.8] to [61.0]. The state's rate now falls below the interval, and the relative standard error drops to about [13] percent. Pooling trades detail about single years for a stable estimate, which is usually the better choice for small populations.

What the Exercise Shows

An interval's width depends on the amount of information behind the estimate: [1,150] survey respondents give a tight interval for a common behavior, while [14] events give a wide one for a rare outcome. Reporting the interval alongside every estimate lets readers judge how much weight it can bear (Rosner, 2016).

Why the County Asked

The county's overdose prevention coalition wanted to know whether its overdose rate was higher than the state's and whether its smoking rate justified a new cessation program. Both are comparison questions, and an estimate without an interval cannot answer either. The coalition's draft report had printed the single-year overdose rate in a headline as nearly twice the state's, which would have been misleading. After seeing the intervals, the coalition rewrote the headline around the five-year rate and moved the single-year figure to a footnote with its reliability warning.

What a Confidence Interval Means

A [95] percent confidence interval comes from a method that, applied to many samples, would capture the true value in [95] percent of them. It is a statement about the method's reliability, not a probability that the true value lies within one particular interval. In everyday use, it marks out which true values the data would not contradict: narrow intervals mean the estimate pins down the true value closely, and wide intervals mean it does not.

Planning the Next Survey

If the county wants a tighter estimate for smoking, halving the margin of error would take roughly four times as many respondents, since precision grows only with the root of the sample. For overdose deaths, no survey can help; only more years of data or a larger population narrow the interval, which is why pooling is the practical choice.

How to Report Each Estimate

Smoking: "[16.8] percent of adults smoke (95 percent confidence interval, [14.6] to [19.0])." Overdose, single year: "[56.9] per [100,000]; this rate is based on fewer than [20] deaths and should be interpreted with caution." Overdose, five years: "[47.2] per [100,000] (95 percent confidence interval, [35.8] to [61.0])."

Common Errors to Avoid

Using the normal approximation for a rate based on a handful of events. Saying an interval has a [95] percent chance of containing the true value after it has been calculated; the correct statement is that the method captures the true value in [95] percent of samples. Comparing a county rate with the state without considering the county's interval.

References

Rosner, B. (2016). Fundamentals of biostatistics (8th ed.). Cengage Learning.

Ulm, K. (1990). A simple method to calculate the confidence interval of a standardized mortality ratio (SMR). American Journal of Epidemiology, 131(2), 373-375. https://doi.org/10.1093/oxfordjournals.aje.a115507

Xu, J., Murphy, S. L., Kochanek, K. D., & Arias, E. (2021). Deaths: Final data for 2019. National Vital Statistics Reports, 70(8). National Center for Health Statistics. https://doi.org/10.15620/cdc:106058