Chapter 7

µ subx =np δ sub x=√np(1-p)

The mean and standard deviation of a binomial random variable.

z=x-µ/δ

z is obtained by "standardizing": subtracting the mean and then dividing by the standard deviation. When x has a normal distribution, z has a standard normal distribution.

µ subx, and δ subx

The mean and standard deviation, respectively, of a random variable x. These quantities describe the center and extent of spread about the center of the variable's probability distribution.

µ subx=∑xp(x)

The⁴ mean value of a discrete random variablex; it locates the center of the variable's probability distribution.

Normal approximation to the binomial distribution

When both np ≥ 10 and n (1-p)≥ 10, binomial probabilities are well approximated by corresponding areas under a normal curve with µ = n p and δ = √(np(1-p)

Normal probability plot

a ( ) is a graph used to judge the plausibility of the assumption that a sample has been selected from a normal population distribution. If the plot is reasonably straight, this assumption is reasonable

Normal Distribution

A continuous probability distribution that symmetric, bell-shaped probability distribution with mean µ and standard deviation δ . If observations follow a normal distribution,the interval ( ± 1) contains 68% of the observations the interval ( ± 2) contains 95% of the observations andthe interval ( ± 3) contains 99.7% of the observations . It is also called the gaussian distribution.

Probability distribution of a discrete random variable x

A formula, table, or graph that gives the probability associated with each possible x value. Conditions on p(x) are (1) p(x)≥0, and (2) ∑p(x)=1, where te sum is over all possible x values.

Random Variable: discrete or continuous

A numerical variable with a value determined by the outcome of a chance experiment. A random variable is discrete if its possible values are isolated points along the number line and continuouis if its possible values form an entire interval on the number line.

Probability distribution of a continuous random variable x

Specified by a smooth (density) curve for which the total are under the curve is 1. The probability P(a<x<b) is the area under the curve and above the interval from a to b; this is also P(a≤x≤b)

Standard Normal Distribution

The standard normal distribution is a normal probability distribution with μ = 0 and σ = 1. The density curve is called the z curve, and z is the letter commonly used to denote a variable having this distribution. Area. The total area under its density curve is equal to 1, but to the left of z denotes the tablulation area. See Table 2 Appendix.

δ² subx=∑(x-µ subx)²p(x) δ subx=√δ²subx

The variance and standard deviation, respectively, of a discrete random variable; tese are measures of teh extent to which the variable's distribution spreads out about the mean µ subx

Binomial probability distribution p(x)=[n! / x!(n-x)! ] *p^×(1-p)ⁿ⁻×

distributions that allow us to deal with circumstances in which the outcomes belong to two relevant categories, such as acceptable/defective. ---To be a binomial probability distribution, all of the following requirements must be met: 1. The procedure has a fixed number of trials. 2. The trials must be independent (The outcome of any individual trial doesn't affect the probabilities in the other trials). 3. Each trial must have all outcomes classified into two categories (often referred to as success and failure). 4. The probability of a success remains the same in all trials. ---Notation for binomial probability distribution: n = the fixed number of trials x = specific number of successes in 'n' trials, so 'x' can be any whole number between 0 and 'n'. p = the probability of success in one of the 'n' trials q = the probability of failure in one of the 'n' trials P(x) = the probability of getting exactly 'x' successes among the 'n' trials P(S) = p .... is the probability of a success P(F) = 1 - p = q ....is the probability of a failure