Methods - Exam 2, Ch. 6-8

Random Variable

NUMERICAL description of the outcome an experiment.

Discrete Random Variable

DEFINED values; finite or infinite range. Ex: Cars are integer values; can never be a decimal.

Continuous Random Variable

assumes ANY numerical value in an interval/collection of intervals.

Probability Distribution

describes how probabilities are distributed over the values of random variable.

2 Types of Discrete Probability Distribution

1) Uses the classical, subjective, or relative frequency method to determine probabilities for each random variable, 2) Uses a special mathematical formula to compute probabilities (uniform, binomial, poisson)

Probability Function

f(x); provides the probability for each random variable. It must be between 0 and 1, and the sum of all probabilities = 1.

Empirical Discrete Distribution

usage of relative frequency method to develop discrete probability distributions.

Discrete Uniform Probability Distribution

f(x) = 1/n where n is the number of different values a random variable can assume; values are EQUALLY LIKELY.

Expected Value

weighted average of the values the random variable can assume; ∑xf(x) where f(x) is the weight

Variance

weighted average of the squared deviations of a random variable from its mean; ∑(x-μ)²f(x) where f(x) is the weight

Standard Deviation

the positive square root of the variance.

Binomial Experiment

n identical trials, 2 outcomes (success, failure), probabilities don't change from trial to trial, and trials are independent.

Binomial Variables

p = probability of success, (1 - p) = probability of failure, x = number of successes, f(x) = probability of successes

Binomial Expected Value

np

Binomial Variance

np(1-p)

Binomial Standard Dev

√np(1-p)

Poisson Probability Distribution

estimates occurrences over a specified interval of time/space; may assume an infinite sequence of values.

Poisson Experiment

probability of occurrences are the same for any 2 equal-length intervals, and the occurrences/non-occurrences are independent of each other among the intervals.

Poisson Variables

x = number of occurrences in an interval, f(x) = probability of occurrences in an interval, μ = mean number of occurrences in an interval

Poisson End

no stated upper limit; stop when x becomes large enough such that f(x) is approximately 0.

Poisson Mean

equal to the variance

Poisson Variance

equal to the mean

Poisson Standard Deviation

square root of the mean/variance

Continuous Random Variable

can assume ANY value in an interval or in a collection of intervals; no probability for a definite value. It's only defined within a given interval.

Continuous Probability Distribution

probability of a random variable assuming a value within a given interval is the area under the graph.

Continuous Uniform Probability Distribution

random variable is uniformly distributed; a straight line across, forming a box.

Continuous Uniform Probability Function

f(x) = 1/(b-a) for a <= x <= b. Outside of [a, b], f(x) = 0

Uniform Expected Value

E(x) = (a+b)/2

Uniform Variance

Var(x) = (b-a)²/ 12

Normal Probability Distribution

most common distribution, symmetric, 0 skewness, total area = 1, probabilities to left of mean = 0.5 and right of mean = 0.5, and empirical rule.

Normal Mean

the highest point on the normal distribution, can be negative, positive, or 0. Also defines the median and mode too.

Normal Standard Deviation

defines the width of the normal distribution. Large = wide, small = thin.

Empirical Rule

68.3% are within +/- 1 standard dev, 95.4% are within +/- 2 standard dev, and 99.7% are within +/- 3 standard dev.

Standard Normal Distribution

when a normal distribution has a mean of 0 and a standard deviation of 1, where z defines the standard normal variable.

z-value

(x - μ)/σ, the number of standard deviations x is from μ

Probability to the Right

1 - probability to the left.

Exponential Probability Distribution

describes the time it takes to complete a task, skewed to the right with a measure of 2.

Exponential Mean

equal to the standard deviation.

Element

entity on which data are collected.

Population

a collection of all elements of interest.

Sample

a subset of the population.

Sampled Population

population from which the sample is drawn.

Frame

a list of the elements that the sample will be selected from.

Point Estimation

use data from the sample to compute a value for a sample statistic that estimates a population parameter. x̄ -> μ, s -> σ, and p̄ -> p.

Target Population

population we want to make inferences about.

Practical Advice

ensure that the targeted population and sampled population are in close agreement.

Statistical Inference

1) Select a simple random sample n from population. 2) Use sample data to find x̄, s, or p̄. 3) Use x̄, s, or p̄ to make inferences about μ, σ, or p.

Sampling Distribution of x̄

probability distribution of all possible values of x̄.

Unbiased Point Estimator

When E(x̄) = μ. *Expected value

Finite Population Treated as Infinite

when n/N <= .05 where n = sample size and N = population size.

Finite Population Correction Factor

√(N-n)/(N-1)

Standard Error of the Mean

σₓ

Sampling Distribution of ̄x̄ Approximation

approximated via a normal distribution when n >= 30. If population is highly skewed/has outliers, use n = 50. If population has a normal distribution, the sampling distribution is normally distributed for any size.

Central Limit Theorem

select random samples n from a population => sampling distribution of x̄ is approximated by a normal distribution as n becomes large.

Sample Size Increases

σₓ decreases, x̄ has less variability, and x̄ is closer to population mean.

Sampling Distribution of p̄

probability distribution of all possible values of p̄.

Expected Value of p̄

E(p̄) = p

Standard Error of Proportion

σp̄

Sampling Distribution of p̄

approximated by a normal distribution if np >= 5 and n(1-p) >= 5. Then, p̄ = x/n.