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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, $\bar{X}$ , and the sample sum, ΣΧ

If the size (n) of the sample is sufficiently large, then $\bar{X}$ ~ N(μ, $\frac{σ}{\sqrt{n}}$ ) and ΣΧ ~ N(nμ, ( $\sqrt{n}$ )(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, $\frac{σ}{\sqrt{n}}$ , 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 = $\frac{1}{m}$ and the standard deviation is σ = $\frac{1}{m}$ . 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 $\bar{x}$ ) is $\bar{x} = \frac{sum of all values in the sample}{number of values in the sample}$ , and the mean for a population (denoted by μ) is $μ = \frac{sum of all values in the population}{number of values in the population}$ .

normal distribution: a continuous random variable (RV) with probability density function (pdf) $f (x) = \frac{1}{σ \sqrt{2 π}} e^{\frac{- {(x - μ)}^{2}}{2 σ^{2}}}$ , 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 $\frac{σ}{\sqrt{n}}$

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 $μ = \frac{a + b}{2}$ and the standard deviation is $σ = \sqrt{\frac{{(b - a)}^{2}}{12}}$ . The probability density function is $f (x) = \frac{1}{b - a}$ for a < x < b or a ≤ x ≤ b. The cumulative distribution is P(X ≤ x) = $\frac{x - a}{b - a}$ .