# Chapter 4. Probability & Related Topics for Making Inferences About Data

- Probability is an important concept in statistics. Both objective and subjective probabilities are used in the medical field.
- Basic definitions include the concept of an event or outcome. A number of essential rules tell us how to combine the probabilities of events.
- Bayes’ theorem relates to the concept of conditional probability—the probability of an outcome depending on an earlier outcome. Bayes’ theorem is part of the reasoning process when interpreting diagnostic procedures.
- Populations are rarely studied; instead, researchers study samples.
- Several methods of sampling are used in medical research; a key issue is that any method should be random.
- When researchers select random samples and then make measurements, the result is a random variable. This process makes statistical tests and inferences possible.
- The binomial distribution is used to determine the probability of yes/no events—the number of times a given outcome occurs in a given number of attempts.
- The Poisson distribution is used to determine the probability of rare events.
- The normal distribution is used to find the probability that an outcome occurs when the observations have a bell-shaped distribution. It is used in many statistical procedures.
- If many random samples are drawn from a population, a statistic, such as the mean, follows a distribution called a sampling distribution.
- The central limit theorem tells us that means of observations, regardless of how they are distributed, begin to follow a normal distribution as the sample size increases. This is one of the reasons the normal distribution is so important in statistics.
- It is important to know the difference between the standard deviation, which describes the spread of individual observations, from the standard error of the mean, which describes the spread of the mean observations.
- One of the purposes of statistics is to use a sample to estimate something about the population. Estimates form the basis of statistical tests.
- Confidence intervals can be formed around an estimate to tell us how much the estimate would vary in repeated samples.

Neisseria meningitidis, a gram-negative diplococcus, has as its natural reservoir the human posterior nasopharynx where it can be cultured from 2–15% of healthy individuals during nonepidemic periods. The bacterial organism can be typed into at least 13 serogroups based on capsular antigens. These serogroups can be further subdivided by antibodies to specific subcapsular membrane proteins. In the United States, sero groups B and C have accounted for 90% of meningococcal meningitis cases in recent decades. The major manifestations of meningo coccal disease are acute septicemia and purulent meningitis. The age-specific attack rate is greatest for children under 5 years of age.

Epidemiologic surveillance data from the state of Oregon detected an increase in the overall incidence rate of meningococcal disease from 2 cases per 100,000 population during 1987–1992 to 4.5 cases per 100,000 population in 1994 (Diermayer et al, 1999). Epidemiologists from Oregon and the Centers for Disease Control wanted to know if the increased numbers ...