Probability Distributions and Sampling Methods
Probability Density Function
A function used to compute probabilities for a continuous random variable. The area under the graph of a probability density function over an interval represents probability.
Uniform Probability Distribution
A continuous probability distribution for which the probability that the random variable will assume a value in any interval is the same for each interval of equal length.
Normal Probability Distribution
A continuous probability distribution. Its probability density function is bell-shaped and determined by its mean (μ) and standard deviation (σ).
Standard Normal Probability Distribution
A normal distribution with a mean of zero and a standard deviation of one.
Continuity Correction Factor
A value of 0.5 that is added to or subtracted from a value of x when the continuous normal distribution is used to approximate the discrete binomial distribution.
Exponential Probability Distribution
A continuous probability distribution that is useful in computing probabilities for the time it takes to complete a task (U = expected value or mean).
Sampled Population
The population from which the sample is taken.
Frame
A listing of the elements the sample will be selected from.
Parameter
A numerical characteristic of a population, such as a population mean (μ), a population standard deviation (σ), a population proportion (p), and so on.
Simple Random Sample
A simple random sample of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected.
Sampling Without Replacement
Once an element has been included in the sample, it is removed from the population and cannot be selected a second time.
Sampling With Replacement
Once an element has been included in the sample, it is returned to the population. A previously selected element can be selected again and therefore may appear in the sample more than once.
Random Sample
A random sample from an infinite population is a sample selected such that the following conditions are satisfied:
- Each element selected comes from the same population.
- Each element is selected independently.
Target Population
The population for which statistical inferences such as point estimates are made. It is important for the target population to correspond as closely as possible to the sampled population.
Sampling Distribution
A probability distribution consisting of all possible values of a sample statistic (E = expected value of x, U = population mean).
Standard Error
The standard deviation of a point estimator.
Central Limit Theorem
A theorem that enables one to use the normal probability distribution to approximate the sampling distribution of x̄ whenever the sample size is large.
Relative Efficiency
Given two unbiased point estimators of the same population parameter, the point estimator with the smaller standard error is more efficient.
Consistency
A property of a point estimator that is present whenever larger sample sizes tend to provide point estimates closer to the population parameter.
Stratified Random Sampling
A probability sampling method in which the population is first divided into strata and a simple random sample is then taken from each stratum.
Cluster Sampling
A probability sampling method in which the population is first divided into clusters and then a simple random sample of the clusters is taken.
Systematic Sampling
A probability sampling method in which we randomly select one of the first k elements and then select every kth element thereafter.
Convenience Sampling
A nonprobability method of sampling whereby elements are selected for the sample on the basis of convenience. Standard deviation: σx = (standard deviation σ of x) (O = of the population) (n = sample size) (N = population size).