PU525 Unit 3 Public Health Probability Problem Set Example

Two Contacts Out of Five, Six Cases in One County and a Reading of 140: A Probability Problem Set Moving From Binomial and Poisson Counts to the Normal Curve and Sampling Distributions, So the Standard Error Is in Place Before Any Hypothesis Test

[Student Name]

Purdue University Global

PU525: Biostatistics for Public Health

Unit 3 Probability Problem Set

[Instructor Name]

[Date]

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

Problem 1: Household Spread

Scenario: a person with a respiratory infection lives with [five] household contacts. The secondary attack rate, the chance each contact becomes infected, is [15] percent. Distribution: binomial, because there is a fixed number of contacts, each either infected or not, with the same probability, assumed independent. Arithmetic: probability of none infected is [0.85] to the fifth power, [0.444]. Probability of exactly one is five times [0.15] times [0.85] to the fourth, [0.392]. Probability of two or more is one minus the sum, [0.165]. Meaning: in about one household in six, two or more contacts will become ill, which helps a health department plan follow-up calls.

Problem 2: A Suspected Cluster

Scenario: a county expects [2.4] childhood cancer cases a year based on state rates; this year it recorded [6]. Distribution: Poisson, for counts of rare events in a fixed population and time. Arithmetic: the probability of [6] or more cases when [2.4] are expected is about [0.036]. Meaning: taken alone, six cases looks unusual, a chance of under four percent.

Problem 3: The Same Question Across Fifty Counties

Scenario: the state has [50] counties of similar size, each expecting [2.4] cases. Arithmetic: the chance that at least one county records [6] or more in a given year is one minus [0.964] to the fiftieth power, about [0.84]; on average, nearly two counties would do so each year by chance. Meaning: a result that looks rare in one county is expected somewhere in the state every year. Six cases in one county is surprising; six cases in some county is almost certain. This is why most investigated cancer clusters do not reveal an environmental cause: when many places are watched, unusual counts appear by chance (Goodman et al., 2012). The cluster is not dismissed, but it is investigated with that context in mind.

Problem 4: The Normal Curve

Scenario: systolic blood pressure among adults in a county is approximately normal, with mean [124] and standard deviation [16] mmHg. Distribution: normal, assumed from survey data. Arithmetic: a reading of [140] or more is one standard deviation above the mean, so the probability is about [0.159]. Meaning: about one adult in six would have a reading of [140] or above.

Problem 5: The Sampling Distribution of a Mean

Scenario: a health fair measures [64] adults. What is the chance their average is [128] or more if the county mean is [124]? Distribution: the sample mean is approximately normal, with standard deviation equal to the standard error, [16] divided by the square root of [64], which is [2]. Arithmetic: [128] is two standard errors above [124], so the probability is about [0.023]. Meaning: a single person at [128] is unremarkable, but a group average that high would be unusual by chance, which is the logic later used in hypothesis testing.

Problem 6: How Sample Size Changes the Standard Error

Scenario: with [256] adults instead of [64], the standard error falls to [1]. Meaning: quadrupling the sample halves the standard error. Precision improves with the square root of sample size, which is why doubling a survey's size does not double its precision (Rosner, 2016).

Checking the Conditions

Every distribution carries assumptions, and the set checks them rather than assuming them. The binomial needs independent contacts with the same probability; in a crowded apartment, one infected contact may infect another, so the true chance of two or more cases would be higher. The Poisson needs events that occur independently at a constant average rate; if cases share an inherited risk or a family exposure, they are not independent. The normal model for blood pressure is reasonable for adults but less so at the extremes, where real readings have a longer right tail than the curve allows. The sampling distribution of a mean is close to normal for samples of [64] even when individual values are skewed, which is why the last two problems can rely on it. Stating these conditions turns a calculation into an argument a reader can check.

What the Set Builds

The first three problems describe counts that public health tracks directly. The last three show that a summary from a sample, such as a mean, has its own distribution with a spread, the standard error, that depends on sample size. That idea underlies every confidence interval and test in the units that follow, where a P value asks how often a result this extreme would arise by chance if nothing were going on (Whitley & Ball, 2002).

Common Errors to Avoid

Using the binomial when contacts are not independent, as in crowded housing where one contact may infect another. Treating a cluster found by searching many places as if it had been predicted in advance. Confusing the standard deviation of individuals with the standard error of a mean.

References

Goodman, M., Naiman, J. S., Goodman, D., & LaKind, J. S. (2012). Cancer clusters in the USA: What do the last twenty years of state and federal investigations tell us? Critical Reviews in Toxicology, 42(6), 474-490. https://doi.org/10.3109/10408444.2012.675315

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

Whitley, E., & Ball, J. (2002). Statistics review 3: Hypothesis testing and P values. Critical Care, 6(3), 222-225. https://doi.org/10.1186/cc1493