Tests and Confidence Intervals for a Difference Between Two Population Means
Motivation Suppose you have a population with known and . You then take a sample (perhaps not randomly) and discover...
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Tue 18 July 2017
Poisson Distribution Tests
For a Poission distribution with large , use (i.e. instead of ). It...
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Tue 18 July 2017
Some Comments on Selecting a Test Procedure
Things you should think about: What are the implications of your choice of ? How much did intuition play a role in deciding...
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Tue 18 July 2017
p-Values
The p-value is the smallest significance level (i.e. ) at which would be rejected. So if , you reject...
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Tue 18 July 2017
Tests Concerning a Population Proportion
Let be the number of successes. If \(n<
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Tue 18 July 2017
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...
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Tue 18 July 2017
Hypotheses and Test Procedures
The null hypothesis () vs the alternative hypothesis (): The null hypothesis is assumed to be true (default). The...
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Sun 16 July 2017
Intervals for Non-Normal Distributions
The one-sample t-distribution confidence interval is robust for small or even moderate departures from normality, unless is very...
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Sun 16 July 2017
A Prediction Interval for a Single Future Value For a Normal Distribution
What if you have observations in a normal distribution and want to predict ? The prediction interval is \(\bar{x}\pm...
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Sun 16 July 2017
Intervals Based on a Normal Population Distribution: The T-Distribution
Say you take a sample where is not large. Then the CLT doesn’t apply. We must then know/assume a distribution. Suppose we know it...
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Sun 16 July 2017
The T-Distribution: The T-Distribution
Let be independent and identically distributed from . Define the following random variable:...
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Sun 16 July 2017
Basic Properties of Confidence Intervals
Assume you have a...
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Sun 16 July 2017
Methods of Point Estimation
The Method...
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Sun 09 July 2017
Some General Concepts of Point Estimation
Notation: means the point estimator of...
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Sat 08 July 2017
The Distribution of a Linear Combination
Let have means and variances...
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Wed 07 June 2017
The Distribution of the Sample Mean and Sum
Let be a random sample from a distribution with mean and standard deviation . Then:...
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Wed 07 June 2017
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...
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Wed 07 June 2017
Expected Values, Covariance and Correlation
Expected Value The expected value of a function is...
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Wed 07 June 2017
Jointly Distributed Random Variables
Probability Given two random variables , the joint pdf is given by . Let be an event. Then the joint...
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Wed 07 June 2017
Probability Plots
Sample Percentiles Calculating sample percentiles is challenging. What is the 23rd percentile of 10 points? One rule: Order the ...
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Tue 06 June 2017
Extreme Value Distribution
For Weibull, let . This has both scale and location parameters. Location is and scale is...
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Tue 06 June 2017
The Beta Distribution
The beta distribution The parameters are , and with . \begin{equation*}...
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Tue 06 June 2017
The Lognormal Distribution
If is a normal distribution, then is log-normal. \begin{equation*} f(x;\mu,\sigma)=\frac{1}{\sqrt{2\pi\sigma}...
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Tue 06 June 2017
The Weibull Distribution
Probability Density Function Let \begin{equation*}...
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Tue 06 June 2017
The Erlang Distribution
If the time between successive events is independent each with an exponential distribution with , then the total time ...
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Tue 06 June 2017
The Chi-Squared Distribution
Chi-squared distribution Probability Density Function The parameter is and it is a positive integer. It is the gamma...
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Tue 06 June 2017
The Exponential Distribution
Exponential distribution Probability Density Function Let \begin{equation*} f(x;\lambda)=\lambda e^{-\lambda x}...
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Tue 06 June 2017
The Gamma Distribution
The problem with the normal distribution is that it is symmetric. The Gamma distribution is useful for skewed distributions....
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Tue 06 June 2017
The Normal Distribution
Probability Distribution Function The parameters are and . \begin{equation*}...
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Tue 06 June 2017
Pareto Distribution
The Pareto distribution is good for approximating income distributions or population sizes. The pdf is given by: \begin{equation*}...
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Tue 06 June 2017
Continuous Random Variables and Probability Distributions
Continuous distributions are given by a probability density function (pdf): \begin{equation*} P\left(a\le X\le...
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Tue 06 June 2017
Zipf Distribution
Suppose you have a library with items and you want to sort them by popularity. The parameters are and . The domain...
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Tue 06 June 2017
Poisson Distribution
is of a Poisson distribution if its pmf is where is 0, 1, 2, etc....
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Thu 18 May 2017
Negative Binomial Distribution
The experiment requires: The trials be independent The outcome is binary (success or failure). The probability of success or failure is...
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Thu 18 May 2017
Hypergeometric Distribution
The hypergeometric experiment requires: The population is finite, with individuals. The outcome of each trial is binary (success...
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Thu 18 May 2017
Binomial Distribution
Bernoulli Distribution A Bernoulli random variable is one whose only possible values are 0 and 1. ...
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Thu 18 May 2017
Discrete Random Variables
Random Variables A discrete random variable is one whose set of possible values is countable. Probability Distributions for Discrete...
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Thu 18 May 2017
Read From UIUC
Read hypothesis testing in Chapter 2 (ML rules). Reliability in Chapter 2
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Thu 18 May 2017
Probability
Sample Spaces and Events The sample space is a set. An event is a subset of the sample space. A simple event is an event...
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Wed 17 May 2017
Measures of Location and Variability
Measures of Location When reporting a sample mean, use one extra significant digit. Sample mean: Population mean: ...
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Sat 13 May 2017
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...
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Sat 13 May 2017
Statistics: Overview
Populations, Samples and Processes A census means you poll every member of the population. Univariate vs bivariate or multivariate:...
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Sat 13 May 2017