Chapter 7 Statistics: The normal probability distribution

Normally distributed, normal probability distribution

A continuous random variable is normally distributed or has a normal probability distribution if its relative frequency histogram has the shape of a normal curve.

Normal probability distribution

A continuous random variable whose relative frequency histogram has the shape of a normal curve.

standard normal curve

A normal distribution with mean of 0 and standard deviation of 1. Probabilities given in a table for values of the standard Normal variable. Symmetrical distribution forming a bell-shaped curve in which the mean, median, and mode are all equal and fall in the exact middle

Normal Probability Density Function

The mathematical expression that describes the shape of the normal probability distribution. Describes a symmetric, bell shaped curve; completely defined by the mean and variance (standard deviation)

Standard normal random variable, Z

standard normal random variable whose mean is 0 and standard deviation is 1.

normal density curve

symmetric about the mean μ; has standard deviation σ; highest point at X = μ; total area under the curve = 1; values of the random variable X on x-axis; probabilities are represented by areas under the curve

Discrete v. Continuous Random Variables

two types of quantitative random variables

Normal curve

used to describe continuous random variables that are said to be normally distributed.

Standardizing a Normal Random Variable

z=(x-μ)/σ

Properties of the Normal Curve

1. The normal curve is symmetric about its mean, μ. 2. Because the mean = median = mode, the normal curve has a single peak and the highest point occurs at x = μ. 3. The normal curve has inflection points at μ - σ & μ + σ. 4. The area of the normal curve is 1. 5. The area under the normal curve to the right of μ equals the area under the left of μ, which equals ½. 6. As x increases without bound (gets larger and larger), the graph approaches but never reaches, the horizontal axis. As x decreases without bound (gets more and more negative), the graph approaches but never reaches, the horizontal axis. 7. The Empirical Rule: Approximately 68% of the area under the normal curve is between x = μ − σ and x = μ + σ; approximately 95% of the area is between x = μ − 2σ and x = μ + 2σ; approximately 99.7% of the area is between x = μ − 3σ and x = μ + 3σ.

Uniform Probability Distribution

If any two intervals of equal length are equally likely, a random variable, X, is said to follow uniform probability distribution

Model

In mathematics, a model is an equation, table, or graph used to describe reality.

Standard normal distribution

Is a special normal distribution. It has a mean μ = 0 and standard deviation σ = 1. A standard normal variable is always denoted by the variable Z so it's often referred to as the "Z distribution." Z is only use to denote a standard normal variable. Z sub alpha denotes the Z-score having area, α specifically to its right under the standard normal curve. (α is usually a very small amount of probability).

Inflection points

Points where the curvature of the graph changes.

Area under a Normal Curve

Suppose that a random variable X is normally distributed with mean μ and standard deviation σ. The area under the normal curve for any interval of values of the random variable X represents either the proportion of the population with the characteristic described by the interval of values or the probability that a randomly selected individual from the population will have the characteristic described by the interval of values.

Normal score

The expected z-score of the data value, assuming that the distribution of the random variable is normal. A normal probability plot is a graph that plots observed data versus normal scores. A normal score is the expected​ z-score of a data​ value, assuming that the distribution of the random variable is normal.

Example Interpreting the Area Under a Normal Curve

The probability that X is less than 2100 (the purple area to the left of the curve) is .3085

Probability Density Function (pdf)

an equation used to compute probabilities of continuous random variables. It must satisfy the following two properties: 1. The total area under the graph of the equation over all possible values of the random variable must equal 1. 2. The height of the graph of the equation must be greater than or equal to 0 for all possible values of the random variable.