Statistics Test #2

classical probability

# of favorable outcomes/total # of possible outcomes

Normal Distribution

-it is bell shaped, symmetrical, asymptotic. -its mean, median, and mode are all equal. -total area under its curve is 1.00 -the location is determined by the mean

discrete random variables

is one which may take on only a countable number of distinct values such as 0,1,2,3,4,........If a random variable can take only a finite number of distinct values, then it must be discrete.

contingency table

is used to classify sample observations according to two or more identifiable characteristics measured.

The sum of all collectively exhaustive and mutually exclusive events is what?

it equals 1.0 or 100%

Empirical Rule

-68 percent of the area under the normal curve is within one standard deviation of the mean. -95 percent is within two standard deviations of the mean. -Practically all is within three standard deviations of the mean.

Uniform Probability Distribution

-The uniform probability distribution is used to describe continuous random variables that appear to have equally likely outcomes over their range of possible values. -This distribution is rectangular in shape and is defined by minimum and maximum values.

tree diagram

-Useful for portraying conditional and joint probabilities. -Particularly useful for analyzing business decisions involving several stages. -A graph that is helpful in organizing calculations that involve several stages. -Each segment in the tree is one stage of the problem. The branches of a tree diagram are weighted by probabilities.

Poisson probability distribution

describes the number of times some event occurs during a specified interval. The interval may be time, distance, area, or volume. The distribution is characterized by the number of times an event happens during some interval. -The probability is proportional to the length of the interval & the intervals are independent. P(x) = (u^x)(e^-u)/x!

collectively exhaustive

if at least one of the events must occur when an experiment is conducted.

mutually exclusive

if the occurrence of any one event means that none of the others can occur at the same time.

independent events

if the occurrence of one event does not affect the occurrence of another.

sampling distribution of the sample mean

is a probability distribution consisting of all possible sample means of a given sample size selected from a population. sum of sample means/total # of samples

experiment

is a process that leads to the occurrence of one and only one of several possible observations.

empirical probability

is based the law of large numbers. The key to establishing probabilities empirically is that more observations will provide a more accurate estimate of the probability. the probability of an event happening is the fraction of the time similar events happened in the past

cumulative binomial probability distributions

it refers to the probability that the binomial's random variable falls within a specified range (e.g., is greater than or equal to a stated lower limit and less than or equal to a stated upper limit).

continuous random sample

one which takes an infinite number of possible values. They're usually measurements like height or weight.

simple random sample

sample selected so that each item/person in the population has the same chance of being selected for the sample

standard error of mean

standard deviation/sqrt of n

event

the collection of one or more outcomes of an experiment.

general rule of multiplication

the conditional probability is required to compute the joint probability of two events that are not independent. P(A and B) = P(A)P(B | A)

outcome

the particular result of an experiment.

subjective probability

the probability of a particular event happening that is assigned by an individual based on whatever information is available.

mean of the uniform distribution

u = (a + b) / 2

mean of Poisson distribution

u = (n)(pi)

complement rule

used to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1. P(A) + P(~A) = 1 or P(A) = 1 - P(~A)

standard deviation of the uniform distribution

σ = Sqrt((b - a)^2 / 12)

special rule of addition

If two events A and B are mutually exclusive, the probability of one or the other event's occurring equals the sum of their probabilities.

exponential distribution

P(arrival time < x) = 1 - e^(rate)(x)

special rule of multiplication

Requires that 2 events must be independent, meaning independent if the occurrence of one has no effect on the probability of the occurrence of the other. P(A and B) = P(A)P(B)

mean of binomial distribution

(n)(pi)

measures of dispersion

Measures the amount of spread in a distribution

uniform distribution

P(x) = 1 / (b - a)

probability a sample mean falls within

(X - u)/(stdrd/sqrt of n) or (X - u)/(s/sqrt of n)

variance of binomial distribution

(n)(pi)(1 - pi)

Variance/Standard Deviation of Discrete Probability Distributions

1) Subtract the mean from each value, and square this difference. 2) Multiply each squared difference by its probability. 3) Sum the resulting products to arrive at the variance. The standard deviation is found by taking the positive square root of the variance. Σ[(x - u)^2P(x)]

Binomial Probability Distribution

1) There are only two possible outcomes on a particular trial of an experiment. 2) The outcomes are mutually exclusive 3) The random variable is the result of counts 4) Each trial is independent of any other trial (nCx)(pi^x(1 - pi)^(n - x)) n: # of trials x: random variable defined as # of successes(0) pi: probability of success on each trial

correction factor

1. For the probability at least X occurs, use the area above (X -.5). 2. For the probability that more than X occurs, use the area above (X+.5). 3. For the probability that X or fewer occurs, use the area below (X -.5). 4. For the probability that fewer than X occurs, use the area below (X+.5).

probability distribution

A listing of all outcomes of an experiment and the probability associated with each outcome. -The probability of a particular outcome is between 0 and 1 inclusive. -The outcomes are mutually exclusive events. -The list is exhaustive. So the sum of the probabilities of the various events is equal to 1.

permutations

A permutation is any arrangement of r objects selected from n possible objects. Permutations are for lists (order matters). n!/(n - r)! n=total # of objects r=# of objects selected

Cluster Sampling

A population is divided into clusters using naturally occurring geographic or other boundaries. Then, clusters are randomly selected and a sample is collected by randomly selecting from each cluster.

Stratified Random Sampling

A population is first divided into subgroups, called strata, and a sample is selected from each stratum. Useful when a population can be clearly divided in groups based on some characteristics

joint probability

A probability that measures the likelihood two or more events will happen concurrently.

probability

A value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur.

expected value

E(X): The mean of a probability distribution. The mean is a typical value used to represent the central location of a probability distribution. Σ[xP(x)]

factorials

Factorials show how many different ways there are to order or arrange a set of things. n! = n (n-1) (n-2) (n-2) ... (3) (2) (1)

general rule of addition

If A and B are two events that are not mutually exclusive, then P(A or B) is given by the following formula: P(A or B) = P(A) + P(B) - P(A and B)

Central Limit Theorem

If all samples of a particular size are selected from any population, the sampling distribution of the sample mean is approximately a normal distribution. This approximation improves with larger samples.

multiplication formula

If there are m ways of doing one thing and n ways of doing another thing, there are m × n ways of doing both.

z-score

The Z-Score is a normalized score from a set of distribution data that tells you how many standard deviations your score is from the mean. You need a z-score to use the z-table

Systematic Random Sampling

The items or individuals of the population are arranged in some order. A random starting point is selected and then every kth member of the population is selected for the sample.

law of large numbers

The more times an experiment is repeated the closer the probability of each outcome gets closer to the actual probability.

Normal Approximation to the Binomial

The normal distribution (a continuous distribution) yields a good approximation of the binomial distribution (a discrete distribution) for large values of n. The normal probability distribution is generally a good approximation to the binomial probability distribution when n and n(1- ) are both greater than 5.

combinations

The number of ways to choose r objects from a group of n objects without regard to order. Combinations are for groups (order doesn't matter). n!/r!(n - r)! n=total # of objects r=# of objects selected

The Standard Normal Probability Distribution ( z distribution)

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. -A z-value is the signed distance between a selected value, designated X, and the population mean , divided by the population standard deviation, σ. z = (x - u)/ σ

Hypergeometric Probability Distribution

The trials are not independent, meaning that the outcome of one trial affects the outcome of any other trial. Use hypergeometric distribution if the experiment is binomial, but sampling is without replacement from a finite population where n/N is more than 0.05. P(x) = ((sCx)(N-sCn-x))/(NCn) N: size of population S: # of successes in the population x: # of successes in sample n: size of the sample or # of trials

conditional probability

There is additional knowledge that can alter the pre-established probabilities. Probability of one event is dependent on the results of the previous event. P( B | A )

random variable

a quantity resulting from an experiment that, by chance, can assume different values. There are two types of random variables, discrete and continuous.

Continuous Probability Distributions

a random variable that can assume any value within some interval or intervals. They have graphs called density functions, where the probability that x falls between two values a and b, is the area under the curve between a and b.

probability sample

a sample selected such that each item or person in the population being studied has a known likelihood of being included in the sample. Simple Random Sample Systematic Random Sampling Stratified Random Sampling Cluster Sampling

count data

a type of data in which the observations can take only the non-negative integer values {0, 1, 2, 3, ...}, and where these integers arise from counting rather than ranking.