P-value
The computed value is known as the p-value
It is a measure of plausibility of the null hypothesis
If the probability is low (low p-value), that implies our hypothesis is not
consistent with the null (the sample mean is extreme if the null were tru

2p factorial experiments
Extending to p-factors
As the number of factors increase, the number of treatment combinations
increases exponentially in the number of treatment levels
Restrict discussion to factors with 2 levels
Low and High
p factors, each wit

Inferences on the Difference between Two Means
Recall from Chapter 4.3
2
If X and Y be independent normal variables, with X ~ N ( X , X ) and Y ~ N ( Y , Y2 ) .
2
Then X Y ~ N ( X Y , X Y2 )
If X has a large sample size nX and Y has a large sample size nY

One-Way ANOVA
Vocabulary:
Factorial Experiment: vary one or more quantities (factors) to determine the
effect on the response
Factors: quantities varied in experiments
Each factor has levels or treatment levels
Response (dependent) variable: y
Experimenta

Two-Way ANOVA
Two factor experiment:
Example:
Rather than one treatment affecting the response, there are 2
Can analyze, not only how they affect the response individually but also how they
jointly affect the response
If responses are observed on every tr

t-test
The method is essentially the same as the hypothesis test for large sample
(chapter 6.2)
With the following modifications:
assume the population distribution is normally distributed because the
sample size is not large enough for the Central Limit

Power
Decisions and consequences
Truth
H0 true
H0 false
Reject H0
Type I error
Correct decision
Fail to reject H0
Correct decision
Type II error
Type I error is measured by (significance)
Recall, we know the probability of correct decision if we reject th

One-Way ANOVA
Vocabulary:
Factorial Experiment: vary one or more quantities (factors) to determine the
effect on the response
Factors: quantities varied in experiments
Each factor has levels or treatment levels
Response (dependent) variable: y
Experimenta

Inferences on the Difference between Two Means
Recall from Chapter 4.3
2
If X and Y be independent normal variables, with X ~ N ( X , X ) and Y ~ N (Y , Y2 ) .
2
Then X Y ~ N ( X Y , X Y2 )
If X has a large sample size nX and Y has a large sample size nY

Basics
Stat 3450 is a course in probability modeling and statistical inference
In this course we will learn how to
- Describe random variation and uncertainty mathematically via probability
- Build probability models to describe the relationship between t

Conditional Probability and Independence
In many experiments, knowing one event has occurred can modify our assignment
of probability to other events.
Example:
Conditional probability is a probability that is based on a part of a sample space.
Uncondition

2p factorial experiments
Extending to p-factors
As the number of factors increase, the number of treatment combinations
increases exponentially in the number of treatment levels
Restrict discussion to factors with 2 levels
Low and High
p factors, each wit

Randomized Complete Block Design
Blocking Factors
Factors that may affect the response but are not of interest to our analysis
are called blocking factors
The blocking factors are introduced to explain some inherent variability in
the response and untangl

Hypothesis tests for population means; large sample
Goal:
Structure of hypothesis testing:
2 hypotheses:
Formulating the hypothesis
Rules
Always place the equality in the null
Put what you want to prove in the alternative
Make sure the hypotheses refer to

Small Sample Confidence Intervals
Recall:
For large sample size (n > 30) according to the Central Limit Theorem
X ~ N ( ,
2
n
)
Furthermore for large n s so
s2
X ~ N ( , )
n
The standardized version:
X
=Z
s
where Z ~ N (0, 1)
n
What if n is small?
If the

t-test
The method is essentially the same as the hypothesis test for large sample
(chapter 6.2)
With the following modifications:
assume the population distribution is normally distributed because the
sample size is not large enough for the Central Limit

P-value
The computed value is known as the p-value
It is a measure of plausibility of the null hypothesis
If the probability is low (low p-value), that implies our hypothesis is not
consistent with the null (the sample mean is extreme if the null were tru

Randomized Complete Block Design
Blocking Factors
Factors that may affect the response but are not of interest to our analysis
are called blocking factors
The blocking factors are introduced to explain some inherent variability in
the response and untangl

Hypothesis tests for population means; large sample
Goal:
Structure of hypothesis testing:
2 hypotheses:
Formulating the hypothesis
Rules
Always place the equality in the null
Put what you want to prove in the alternative
Make sure the hypotheses refer to

Small Sample Confidence Intervals
Recall:
For large sample size (n > 30) according to the Central Limit Theorem
X ~ N ( ,
2
n
)
Furthermore for large n s so
s2
X ~ N ( , )
n
The standardized version:
X
=Z
s
where Z ~ N (0, 1)
n
What if n is small?
If the

Fractional factorial experiments
Why fractional?
As the number of factors increases, the requisite number of observations needed
goes up very fast.
Sometimes you cannot afford enough observations.
Fewer observations than effects/interactions lead to confo

Power
Decisions and consequences
Truth
H0 true
H0 false
Reject H0
Type I error
Correct decision
Fail to reject H0
Correct decision
Type II error
Type I error is measured by (significance)
Recall, we know the probability of correct decision if we reject th

Two-Way ANOVA
Two factor experiment:
Example:
Rather than one treatment affecting the response, there are 2
Can analyze, not only how they affect the response individually but also how they
jointly affect the response
If responses are observed on every tr

Probability
Definition:
An experiment is a process with an outcome that cannot be predicted ahead of
time with certainty.
Examples:
Definition:
A sample space is the set of all possible outcomes.
Examples:
Notation: set notation.
Set operations:
Union
Int

Discrete Random Variables
In most applications we are usually interested in one or more real-valued
summaries of the outcome of the experiment.
A random variable assigns a numerical value to each outcome in a sample space.
It is denoted usually by capital

Discrete Random Variables
In most applications we are usually interested in one or more real-valued
summaries of the outcome of the experiment.
A random variable assigns a numerical value to each outcome in a sample space.
It is denoted usually by capital

Normal Distribution
The most important and widely-used probability distribution in statistics.
Used as model for many physical measurements like heights, weights, test scores,
errors in measurement, etc.
Central Limit Theorem
Under certain conditions sums

Random Samples
First take a look at the functions of random variables
Example:
If salaries in a company have mean $25000 and standard deviation $10000.
How do these summary statistics change (or not) if everyone in the company is
given a $500 bonus?
Mean:

More on Continuous Random Variables including Mean, Variance, and Standard
Deviation
Recall: Probabilities compute using definite integrals. If f (x) is the probability
density function and F(x) is the cumulative distributive function of a continuous
rand

Binomial Distribution
Probability distributions (probability mass functions or probability density
functions) for random variables can be derived from simple experiments.
Simple examples:
flipping coins, rolling dice, dealing cards, etc.
In many cases, sc