Sections
Key Terms
Key Terms
- average
- a number that describes the central tendency of the data; there are a number of specialized averages, including the arithmetic mean, weighted mean, median, mode, and geometric mean
- central limit theorem
- given a random variable (RV) with a known mean, μ, and known standard deviation, σ, and sampling with size n, we are interested in two new RVs: the sample mean, , and the sample sum, ΣΧ
If the size (n) of the sample is sufficiently large, then ~ N(μ, ) and ΣΧ ~ N(nμ, ()(σ)). If the size (n) of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The mean of the sample means will equal the population mean, and the mean of the sample sums will equal n times the population mean. The standard deviation of the distribution of the sample means, , is called the standard error of the mean.
- exponential distribution
- a continuous random variable (RV) that appears when we are interested in the intervals of time between a random events; for example, the length of time between emergency arrivals at a hospital, notation: X ~ Exp(m)
The mean is μ = and the standard deviation is σ = . The probability density function is f(x) = me–mx, x ≥ 0, and the cumulative distribution function is P(X ≤ x) = 1 – e–mx.
- mean
- a number that measures the central tendency; a common name for mean is average; the term mean is a shortened form of arithmetic mean;
by definition, the mean for a sample (denoted by ) is , and the mean for a population (denoted by μ) is .
- normal distribution
- a continuous random variable (RV) with probability density function (pdf) , where μ is the mean of the distribution and σ is the standard deviation; notation: Χ ~ N(μ, σ). If μ = 0 and σ = 1, the RV is called a standard normal distribution
- sampling distribution
- given simple random samples of size n from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution
- standard error of the mean
- the standard deviation of the distribution of the sample means, or
- uniform distribution
- a continuous random variable (RV) that has equally likely outcomes over the domain a < x < b; often referred as the rectangular distribution because the graph of the pdf has the form of a rectangle
Notation: X ~ U(a, b). The mean is and the standard deviation is . The probability density function is for a < x < b or a ≤ x ≤ b. The cumulative distribution is P(X ≤ x) = .