Understanding Probability Distributions: Random Variables, Discrete & Continuous, Binomial, Study notes of Statistics

Download Understanding Probability Distributions: Random Variables, Discrete & Continuous, Binomial and more Study notes Statistics in PDF only on Docsity! Chapter 6 Probability Distributions 6.0 Probability Basics A. Random Phenomenon/Activity An activity with possible outcomes; cannot predict with certainty which outcome will result. B. Sample Space The set of possible outcomes of a random activity C. Probability of an Outcome in a Sample Space. A measure of the likelihood (0 - 1 or 0% - 100%) that the outcome will result when the activity is performed. Larger values mean greater likelihood; smaller values mean smaller likelihood. D. Examples • – Random Activity: Toss a fair coin - observe upface – Sample Space = {H,T } – Assuming fair coin, P[H] = 0.5, P[T] = 0.5 – If outcomes in sample space are equally likely, the probabilities are identical, equal to 1 di- vided by number of outcomes 1 2 • – Random Activity: Toss a fair die - observe number of dots on upface. – Sample Space = {1,2,3,4,5,6} – Assuming fair die, P[1] = P[2] = P[3] = P[4] = P[5] = P[6] = 1/6 • – Random Activity: Drop a thumbtack - ob- serve whether point land up or down – Sample Space = {up, down} – Probabilities: Cannot get by equally likely ar- gument, must drop thumback large number of times. Then P[up] = fraction of times point lands up, P[down] = fraction of times lands down. Here probabilities are obtained by observa- tion, repeating random activity large number of times. • – Random Activity: Select JMU student at ran- dom and determine if student is vegetarian (V) or not (NV). – Sample Space = {V,NV} – Outcomes V,NV not equally likely. P[V] = fraction of vegetarians in population, unknown. P[NV] = fraction of non-vegetarians in popu- lation. – When selecting a simple random sample of size n = 1 from a population, then proba- bilities of possible outcomes are fractions in population 5 • The standard deviation σ is the standard devi- ation of a large number of observations of the random variable X – thus it is referred to as a population standard deviation. The standard deviation s = √ Σ(x−x)2 n−1 from Chapter 2 would be the standard deviation of a small number or sam- ple of observations on the random variable X – thus it is now called a sample standard devia- tion. D. Probability Distributions for Continuous Random Vari- ables • Page 253. A continuous random variable has possible values that form an interval. Its prob- ability distribution is specified by a curve that determines the probabilities that the random vari- able falls in any particular interval of values: – Each interval has probability between 0 and 1. This is the area under the curve, above that interval – The interval containing all possible values has probability equal to 1, so the total area under the curve equals to 1. • Example 6 6.2 How Can We Find Probabilities for Bell Shaped Distributions A. The Normal Probability Distribution • Symmetric, bell-shaped, characterized by its mean µ and standard deviation σ. • Probability within any particular number of stan- dard deviations of µ is the same for all normal distributions (0.68 within 1 standard deviation, 0.95 within 2 standard deviations, and 0.997 within 3 standard deviations • Standard Normal Probability Distribution - nor- mal probability of z -scores, where z = (x−µ)/σ; B. Finding probabilities about z-scores - Table A, page A1, A2 in the appendix C. Finding probabilities about more general normal ran- dom variables, X • Examples 7 6.3 How Can We Find Probabilities When Each Observation Has Two Possile Outcomes? A. The Binomial Distribution: Probabilities for Counts with Binary Data • Binary observation Observation that takes on one of two possible outcomes • Random phenomenon: observing n cases or trials of binary observations Summary = number or proportion of n binary cases or trials with outcome of interest. Under certain CONDITIONS the number with outcome of interest has the binomial distribution. • Binomial Conditions – Random phenomenon has n trials, each with two possible outcomes. Outcome of interest is called a “success” and the other outcome is called a “failure”. – Each trial has the same probability of success, denoted by p. – The probability of failure is denoted by 1− p. – The n trials are independent. That is, the result for one trial does not depend on the results of other trials. • Binomial Random Variable X = number of suc- cesses in the n trials for binomial conditions. • Example - Binomial conditions, Binomial Ran- dom Variable 10 • Sampling distribution of sample proportion is ap- proximately normally distributed if np ≥ 15 and n(1 − p) ≥ 15 Example: D. Standard Error • The standard deviation of a sampling distribu- tion is called a standard error 6.5 How Close are Sample Means to Population Means? A. The Sampling Distribution of the Sample Mean • Sample Mean X as a (random) variable • Sampling Distribution of the Sample Mean X. Example (when population sampled is small) 11 B. Mean and Standard Deviation of the Sampling Dis- tribution of X • Suppose random sampling from a population with mean µ and and standard deviation σ. • Mean of the sampling distribution of X is equal to population mean µ. • Standard Deviation of the sampling distribution of X is σ/ √ n (assumes sample size is small rela- tive to pop. size) C. Getting Probabilities about X • Normal Populations If the population is normally distributed, then the sampling distribution of X is normal, regard- less of the size of n • Non-normal populations (Central Limit Theorem) Even if the population is not normal, then the sampling distribution of X is approximately nor- mal for “large” n (Rule of Thumb: n ≥ 30) Example: 6.58, page 297 6.6 How Can We Make Inferences about a Pop- ulation? A. Three Types of Distributions • Population Distribution - see page 298 • Data Distribution - see page 298 • Sampling Distribution - see page 298 12 B. Examples of Three Types of Distributions • Categorical Population (S/F), see Example 22 • Quantitative Population C. Standard Error used to determine how close sample prediction is to population parameter