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z Tests and Confidence Intervals for a Difference Between Two Population Means

Motivation Suppose you have a population with known μ and σ. You then take a sample (perhaps not randomly) and discover...

Poisson Distribution Tests

For a Poission distribution with large n, use Z=X¯λλ/n (i.e. λ instead of S). It...

Some Comments on Selecting a Test Procedure

Things you should think about: What are the implications of your choice of α? How much did intuition play a role in deciding...

p-Values

The p-value is the smallest significance level (i.e. α) at which H0 would be rejected. So if pα, you reject...

Tests Concerning a Population Proportion

Let X be the number of successes. If \(n<

Tests About a Population Mean

Steps to Carry Out The Experiment Identify the parameter of interest. Determine the null value and state the null hypothesis. State the...

Hypotheses and Test Procedures

The null hypothesis (H0) vs the alternative hypothesis (Ha): The null hypothesis is assumed to be true (default). The...

A Confidence Interval For The Median

If X1,,Xn be...

Confidence Intervals For The Variance and Standard Deviation of a Normal Distribution

Let X1,,Xn be...

Intervals for Non-Normal Distributions

The one-sample t-distribution confidence interval is robust for small or even moderate departures from normality, unless n is very...

A Prediction Interval for a Single Future Value For a Normal Distribution

What if you have n observations in a normal distribution and want to predict Xn+1? The prediction interval is \(\bar{x}\pm...

Intervals Based on a Normal Population Distribution: The T-Distribution

Say you take a sample where n is not large. Then the CLT doesn’t apply. We must then know/assume a distribution. Suppose we know it...

The T-Distribution: The T-Distribution

Let X1,,Xn be independent and identically distributed from N(μ,σ2). Define the following random variable:...

Large Sample Confidence Intervals for a Population Mean and Proportion

For any...

Basic Properties of Confidence Intervals

Assume you have a...

Estimating the Mean of a Symmetric Distribution

A good estimate...

Methods of Point Estimation

The Method...

Some General Concepts of Point Estimation

Notation: μ^=X¯ means the point estimator of...

The Distribution of a Linear Combination

Let X1Xn have means μi and variances...

The Distribution of the Sample Mean and Sum

Let X1Xn be a random sample from a distribution with mean μ and standard deviation σ. Then:...

Statistics and Their Distributions

When we take a sample and calculate its mean and standard deviation, this is treated as a random variable for the population...

Expected Values, Covariance and Correlation

Expected Value The expected value of a function h(X,Y) is...

Jointly Distributed Random Variables

Probability Given two random variables X,Y, the joint pdf is given by p(x,y)=P(X=x,Y=y). Let A be an event. Then the joint...

Probability Plots

Sample Percentiles Calculating sample percentiles is challenging. What is the 23rd percentile of 10 points? One rule: Order the n...

Extreme Value Distribution

For Weibull, let Y=ln(X). This has both scale and location parameters. Location is θ1=ln(β) and scale is...

The Beta Distribution

The beta distribution The parameters are α,β>0, and A,B with BA. \begin{equation*}...

The Lognormal Distribution

If Y=lnX is a normal distribution, then X is log-normal. \begin{equation*} f(x;\mu,\sigma)=\frac{1}{\sqrt{2\pi\sigma}...

The Weibull Distribution

Probability Density Function Let α,β>0 \begin{equation*}...

The Erlang Distribution

If the time between successive events is independent each with an exponential distribution with λ, then the total time X...

The Chi-Squared Distribution

Chi-squared distribution Probability Density Function The parameter is ν and it is a positive integer. It is the gamma...

The Exponential Distribution

Exponential distribution Probability Density Function Let λ>0 \begin{equation*} f(x;\lambda)=\lambda e^{-\lambda x}...

The Gamma Distribution

The problem with the normal distribution is that it is symmetric. The Gamma distribution is useful for skewed distributions....

The Normal Distribution

Probability Distribution Function The parameters are <μ< and σ>0. \begin{equation*}...

Pareto Distribution

The Pareto distribution is good for approximating income distributions or population sizes. The pdf is given by: \begin{equation*}...

Continuous Random Variables and Probability Distributions

Continuous distributions are given by a probability density function (pdf): \begin{equation*} P\left(a\le X\le...

Zipf Distribution

Suppose you have a library with M items and you want to sort them by popularity. The parameters are M and α. The domain...

Poisson Distribution

X is of a Poisson distribution if its pmf is p(x;λ)=eλλxx! where x is 0, 1, 2, etc....

Negative Binomial Distribution

The experiment requires: The trials be independent The outcome is binary (success or failure). The probability of success or failure is...

Hypergeometric Distribution

The hypergeometric experiment requires: The population is finite, with N individuals. The outcome of each trial is binary (success...

Binomial Distribution

Bernoulli Distribution A Bernoulli random variable is one whose only possible values are 0 and 1. P(X=1)=p E[X]=p,V[X]=p(1p)...

Discrete Random Variables

Random Variables A discrete random variable is one whose set of possible values is countable. Probability Distributions for Discrete...

Read From UIUC

Read hypothesis testing in Chapter 2 (ML rules). Reliability in Chapter 2

Probability

Sample Spaces and Events The sample space S is a set. An event is a subset of the sample space. A simple event is an event...

Measures of Location and Variability

Measures of Location When reporting a sample mean, use one extra significant digit. Sample mean: x¯ Population mean: μ...

Pictorial and Tabular Methods in Descriptive Statistics

Pictorial and Tabular Methods in Descriptive Statistics Stem and Leaf Display Stem and leaf display Advantages: Useful for displaying...

Statistics: Overview

Populations, Samples and Processes A census means you poll every member of the population. Univariate vs bivariate or multivariate:...