MPH 550 Week 2 Probability and Sampling Example

One in Eight Adults: Using Probability and Sampling Theory to Plan a County Diabetes Survey and Interpret a Screening Test

[Student Name]

University of Phoenix

MPH/550: Public Health Statistics

Week 2 Assignment

[Instructor Name]

[Date]

The health department, its planned survey, screening program and the test accuracy figures are composites or illustrative values written for a model paper; the county estimate comes from CDC PLACES, and other findings come from the sources cited.

The county health department planned two projects for the coming year: a household health survey to measure diabetes and its risk factors directly, instead of relying on modeled estimates, and a free diabetes screening event at a community center. Both depended on probability. This paper applies probability rules, distributions and sampling theory to planning them.

The Starting Probability

According to CDC's PLACES estimates for 2023, about 12.3% of Pueblo County adults have diagnosed diabetes, crude prevalence, with a confidence interval of 10.6% to 14.1% (Centers for Disease Control and Prevention [CDC], 2025). Treated as a probability, if one adult is chosen at random, the chance that person has diagnosed diabetes is about 0.123, and the chance they do not is 0.877.

The Complement Rule

Probabilities of an event and its complement sum to one. That rule answers a practical question: how likely is it that at least one of 10 randomly chosen adults has diagnosed diabetes? The probability that none do, assuming independence, is 0.877 raised to the 10th power, about 0.27. So the probability that at least one does is about 0.73.

The Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent trials with the same probability. For 10 adults, the expected number with diagnosed diabetes is 10 times 0.123, or about 1.2. The probability of finding zero or one is about 65%, and the probability of finding three or more is about 12%. A screener meeting groups of 10 should usually expect one or two people with diabetes, occasionally more.

An Assumption to Check

The binomial model assumes independence. At a community event, people arrive with family members who share diet, genes and neighborhoods, so their diabetes status is not independent. Clustering would make the true spread wider than the binomial predicts.

Conditional Probability and Screening

The screening event would test people without known diabetes. Among them, the department assumed, for planning purposes, that about 3% have undiagnosed diabetes. Suppose the screening test has a sensitivity of 80%, detecting 80% of people who have the disease, and a specificity of 95%, correctly clearing 95% of those who do not. These are illustrative values, not the properties of a particular test.

Bayes' Theorem in Numbers

Picture 1,000 people screened. About 30 have undiagnosed diabetes, and the test finds 24 of them. Of the 970 without diabetes, 5%, or about 48, test positive falsely. Of the 72 positive results, only 24 are true. The positive predictive value is 24 divided by 72, about 33%. The negative predictive value, by contrast, is above 99%. Two of every three positive screens would belong to people without diabetes.

What That Means for the Event

The calculation changed the plan. Staff will describe a positive screen as a reason for a confirmatory test, not a diagnosis, and the department arranged for confirmatory testing through community clinics so positives do not become worry without follow-up.

From People to Samples

The household survey raises a different question: how precisely can a sample estimate the county's prevalence? If many random samples of 400 adults were drawn, their prevalence estimates would vary around the true value. That spread is the sampling distribution, and its standard deviation is the standard error.

Standard Error of a Proportion

The standard error of a proportion comes from the product of the proportion and its complement, divided by the sample size, under a square root. With p of 0.123 and n of 400, it is about 0.016, or 1.6 percentage points. By the normal approximation, about 95% of samples would produce estimates within 1.96 standard errors of the truth, roughly 9.1% to 15.5% if the true value were 12.3%.

Choosing a Sample Size

To estimate prevalence within 3 percentage points with 95% confidence, the required sample is 1.96 squared times p times one minus p, divided by 0.03 squared, about 460 completed interviews. Because the survey will sample households in clusters and weight results, which enlarges variance, staff applied a design effect of 1.5, raising the target to about 690.

Sampling Methods Compared

Simple random sampling gives every adult an equal chance but needs a complete list. Systematic sampling picks every kth unit from a list, which is easy but risky if the list has a pattern. Stratified sampling divides the population into groups, such as neighborhoods, and samples each, guaranteeing representation and improving precision. Cluster sampling selects groups, such as blocks, and interviews within them, reducing travel cost but widening error. The department chose stratified sampling by neighborhood, with extra sampling in east side neighborhoods to allow separate estimates.

Random Error Versus Bias

Sample size controls random error but not bias. If the sampling frame misses groups, or if people who respond differ from those who do not, a large sample will be precisely wrong.

What Nonresponse Research Shows

A review of 30 studies found that a survey's response rate by itself was a poor predictor of nonresponse bias; bias varied more across estimates within the same survey than across surveys, depending on whether the survey topic related to who chose to respond (Groves, 2006). A low response rate is a warning, not a verdict, and a high one is not a guarantee.

Choosing the Frame

Telephone surveys once reached nearly all households through landlines. A 2005 pilot within the BRFSS compared random-digit dialing with a mail survey sent to a random sample of residential addresses and found that the mail approach could achieve higher response rates in states where telephone response was below 40%, reached households with only cell phones and offered cost savings (Link et al., 2008). The department adopted address-based sampling, with mail invitations, a web option and follow-up by phone.

Reaching Spanish-Speaking Households

About two in five county residents are Hispanic, and some older adults speak mainly Spanish. The survey will provide every mailed invitation, questionnaire and phone follow-up in both English and Spanish, and bilingual staff will field calls. Without that step, the frame would reach these households but the instrument would not, producing nonresponse concentrated in one group whose diabetes risk is higher.

Incentives and Follow-Up

To improve response, the department will send a prepaid five-dollar incentive with the first mailing, a reminder postcard after one week and a replacement questionnaire after three weeks. Households that do not respond by mail or web will receive up to three phone attempts where a number can be matched to the address.

Weighting

After collection, responses will be weighted to match the county's census distribution by age, sex, ethnicity and neighborhood, reducing bias from groups that respond less. Weighting helps only for differences captured by those variables.

Limits

The screening calculations depend on assumed test accuracy and undiagnosed prevalence. The sample size assumes the survey achieves about 690 completed interviews, and a lower number would widen margins. Language barriers and distrust could reduce response among the residents the department most needs to hear from.

Conclusion

Probability turned a 12.3% estimate into planning tools: what a screener will encounter, what a positive test really means and how many survey responses are needed. The sampling discussion showed that precision is only half of accuracy; frames and nonresponse determine the rest. Evidence on nonresponse bias and address-based sampling shaped a survey design suited to the county.

References

Centers for Disease Control and Prevention. (2025). PLACES: Local data for better health, county data, 2025 release [Data set]. https://data.cdc.gov/d/swc5-untb

Groves, R. M. (2006). Nonresponse rates and nonresponse bias in household surveys. Public Opinion Quarterly, 70(5), 646-675. https://doi.org/10.1093/poq/nfl033

Link, M. W., Battaglia, M. P., Frankel, M. R., Osborn, L., & Mokdad, A. H. (2008). A comparison of address-based sampling (ABS) versus random-digit dialing (RDD) for general population surveys. Public Opinion Quarterly, 72(1), 6-27. https://doi.org/10.1093/poq/nfn003