Probability Distributions and Sampling Statistics Essentials

Quantitative Variables

1. Discrete: values of x have "gaps" between them (ex: test scores)

2. Continuous: no gaps between x values

Random Variable X

Assigns a number to outcomes of an experiment

Probability Distribution of a Discrete X

A table, graph, or forumla that shows the values of X and P(X) where:

Expected Value of Discrete X

1. The probability distribution of x represents a population (μ,σ^2,σ)

2. Mean of X = expected value of X

(1) Discrete in General

Variance of Discrete X

Standard Deviation of Discrete X

Ex:

(2) Binomial X

• Need n, P and q
• x = # of successes in n
• P(X) use binomial tables (if n < 5) or="" the="">

• mean --> μ = nP   variance --> σ^2 = nPq   standard deviation --> σ = √nPq
• NOTE: Formula gives P(X=K) but table gives P(X<>
• IF >= or >  you need to use    1 - whatever found in table (THINK ABOUT THIS)

(3) Poisson X

• Need the mean of x = λ
• x = # of times an event occurs in unit of time or space
• Use poisson tables to get P(X)
• μ = λ   σ^2 = λ   σ = √λ

(4) Uniform X

• x is uniform from c to d (given)
• Draw rectangle and find the height
• P(X) is the area of rectangle
• 

(5) Standard Normal Z

• Stated as standard normal Z
• μ = 0 and σ = 1
• Find P(Z) using normal tables
• Find values of Z using normal tables

(6) Normal X

• Normal is stated and μ and σ given
• Find P(X) using normal tables
• Find values of X using normal tables
• Is it X or Xbar
• 

(7) Xbar = Sample Mean

• Describe the distribution of Xbar
• Find P(X) ----->  which is

Definitions:

• Parameter - number that describes the population
  • μ, σ^2, σ, P
• Statistic - number that describes the sample
  • Xbar, s^2, s

The Central Limit Theorem

Given any shape population, the sampling disribution of Xbar will be approximately equal to normal if n is large (n>=30)

Note: IF popluation is normal, Xbar is normal for any size n.

Sampling Distribution of Xbar (only applies to Xbar)

Given a population of X's with mean μ and standard deviation σ, describe the distribution of Xbar.

• Describe the distribution of Xbar
• Find P(X) -----> 
  •  

If X is normal, use z tables!

E(X) = μ