The Role of Probability and Sampling Distributions

Flashcards covering basic concepts of probability, sampling techniques, probability distributions (Binomial, Normal), evaluating screening tests, independence, Bayes' Theorem, random variables, and sampling distributions.

Probability sampling

All subjects in the population have equal chances of being selected into a sample.

Nonprobability sampling

A sample is selected through a non-random process that does not guarantee equal chances for each subject in the population.

Simple Random Sampling

Each member of the population has an equal chance of being selected.

Systematic Sampling

Subjects are selected by using every kth number after the first subject is chosen at random from 1 through k, where k is calculated as population size / sample size.

Stratified Sampling

The population is first divided into subpopulations, called strata, according to some characteristic important to the study, and then sampling is done from each stratum.

Clustered Sampling

The population is divided into groups called clusters, and a random sample of these clusters is selected. All members of the selected clusters are included in the sample.

Convenience Sampling

Individuals are selected because they are the easiest for the researcher to get information from.

Sample Space

The collection of all possible outcomes, denoted as S.

Event

A specific collection of outcomes, that is, a subset of the sample space.

Probability

The likelihood/chance of an event occurring.

Equally Likely Outcomes

If an experiment consists of N outcomes, and all N outcomes have the same chance of occurring, the probability of each outcome is 1/N.

Conditional Probability

The probability that an event A occurs given that another event B has already occurred, denoted as P(A|B).

Screening tests

Tests commonly used in clinical practice to assess the probability that an individual has a particular disease or medical condition, typically not diagnosing the illness directly.

Sensitivity

The true positive fraction, representing the probability that a person with the disease tests positive, calculated as a/(a+c).

Specificity

The true negative fraction, representing the probability that a person without the disease tests negative, calculated as d/(b+d).

False positive fraction

The probability of testing positive given the absence of disease, calculated as b/(b+d) or 1 – specificity.

False negative fraction

The probability of testing negative given the presence of disease, calculated as c/(a+c) or 1 – sensitivity.

Positive predictive value (PPV)

The percent of positive screen tests that are truly positive, calculated as a/(a+b).

Negative predictive value (NPV)

The percent of negative screen tests that are truly negative, calculated as d/(c+d).

Independent Events

Two events A and B are independent if the occurrence of A does not affect the probability of B occurring, meaning P(B|A) = P(B).

Bayes’ Theorem

A mathematical formula used to compute the conditional probability of events.

Random variable

A function that assigns a numerical value to each outcome in the sample space, serving as a numerical summary of an uncertain outcome from an experiment.

Discrete random variable

A random variable whose set of possible outcomes is countable.

Continuous random variable

A random variable that can take on values on a continuous scale.

Probability distribution

A mathematical description that gives all possible values of a random variable and their associated probabilities.

Bernoulli trial

A random experiment that has only two outcomes (success or failure) or outcomes that can be reduced to two.

Binomial experiment

A sequence of n identical Bernoulli trials where the probability of success remains the same from trial to trial and the trials are independent.

Binomial random variable

The number of successes in n Bernoulli trials.

Binomial distribution

The probability distribution of a binomial random variable.

Normal Distribution

A bell-shaped probability distribution (or bell curve) of a normal random variable.

Empirical Rule (68-95-99.7 Rule)

States that approximately 68% of the values in a normal distribution fall within one standard deviation of the mean, 95% within two, and 99.7% within three.

Standard Normal Distribution

A normal random variable with a mean (µ) of 0 and a standard deviation (σ) of 1, denoted by z.

Percentile

The rth percentile of a distribution is a number such that r% of the data values fall below it and (100 - r)% fall above it.

Parameter

A descriptive measure for a population.

Statistic

A descriptive measure for a sample.

Sampling variability

The variability of a statistic from sample to sample, as its value depends on the particular sample selected from the population.

Sampling distribution

The distribution of the value of a sample statistic for every possible sample of a given size from a population.

Central Limit Theorem (CLT)

For a sufficiently large sample size, the sampling distribution of the sample mean taken from a population with mean µ and standard deviation σ is approximately normal with mean µ and standard deviation σ/√n.

Standard Error (SE)

The standard deviation of the sampling distribution of the sample mean, calculated as σ/√n.