Week 02
Probability Concepts -Review
Probability Concepts - Review
Probability Concepts - Review
Probability Concepts
Review
Multiplication Principle
n2
n1
P
n1
n1
n1
n2
n2
n2
If procedure P can be performed
by n1 ways, and each way is
followed by n2 ways

Lecture 4-3
Central Limit Theorem
Navidi Section 4.11
1
The Central Limit Thereom
The Central Limit Theorem
Let X1,Xn be a random sample from a population with mean and variance
2 .
2
X ~ N ,
n
So the sample means will be normally distributed if the samp

Lecture 8-3
Linear Regression using Excel,
Minitab
1
Regression in Excel
You can use the trendline function to get the
coefficients and R2
This is not enough!
To do a full regression analysis, you need to
use the regression tool under the Data
Analysis

Lecture 2-1:
Probability Concepts
Navidi Chapter 2.1
1
Basic Ideas
Definition: An experiment is a process that results in
an outcome that cannot be predicted in advance
with certainty.
Examples:
rolling a die
tossing a coin
weighing the contents of a box

Lecture 3-2
Propagation of Error Formulas
Navidi Chapter 3.2-3.4
1
Linear Combinations of Measurements
If X is a measurement and c is a constant, then
cX c X .
If X1, Xn are independent measurements and
c1, , cn are constants, then
c1 X1 . cn X n c12

Lecture 5-3
Confidence Intervals for the
Difference in Two Means
Navidi Chapter 5.4, 5.6
1
CI for the Difference in Two Means
Set-Up:
Let X and Y be independent, with X ~ N(X, X2 )
2
and Y ~ N(Y, Y ). Then
X + Y ~ N(X+Y ,
X - Y ~ N(X-Y ,
X2 ) Y2
X2 ). Y

Lecture 2-4:
Linear Functions of Random Variables
Navidi Chapter 2.5
1
Linear Functions of Random Variables
If X is a random variable, and a and b are
constants, then
aX b a X ,b
2
2 2
aX
a
X,
b
aX b a X
.
2
More Linear Functions
If X and Y are random

Lecture 8-1
Correlation
Navidi Chapter 7.1
1
Example
This is a plot of height vs. forearm length for men.
We say that there is a positive association between height and
forearm length. This is because the plot indicates that taller men
tend to have long

Lecture 4-1
Normal Distribution
Navidi Section 4.5
1
Binomial Distribution (4.2)
Poisson Distribution (4.3)
Exponential Distribution (4.7)
Uniform Distribution (4.8)
2
The Normal Distribution
The normal distribution (also called the Gaussian
distribution)

Lecture 6-2
Small Sample Hypothesis Testing
Navidi Chapter 6
1
Small Sample Test for a Population Mean
When we had a large sample we used the sample
standard deviation s to approximate the population
deviation .
When the sample size is small, s may not

Lecture 5-1
Large Sample Confidence Intervals
Navidi Chapter 5.1
1
Introduction
We have discussed point estimates
as an estimate of population mean,
These point estimates are almost never exactly equal
to the true values they are estimating.
In order

Lecture 7-3
Power Calculations
Navidi Chapter 6
1
Errors
When conducting a fixed-level test at significance
level , there are two types of errors that can be
made. These are
Type I error: Reject H0 when it is true.
Type II error: Fail to reject H0 when it

Lecture 5-4
Confidence Intervals for Paired Data
Navidi Chapter 5.7
1
CI for Paired Data
Consider paired data. An example is tread wear on tires. Suppose there
are two different brands on the front wheel of a car. After the car has
been driven for 20,000,

Homework 02
A metallurgist is designing an experiment to determine the effect of flux,
base metal, and energy input on the hardness of a weld. She wants to study
four different fluxes, two different base metals, and three different amounts
of energy input

Week 03
Propagation of Error - Review
FACT: any measuring procedure contains error!
Majority of scientific work is based on calculated values.
Calculated values are obtained from experimental measured
variables.
Error associated with measured quantities p

Lecture 1-1
Course Introduction
1
Instructors
Dr. Brian Tande
Associate Professor and Chair
Harrington Hall Room 323
Phone: 701-777-2337
Email:brian.tande@engr.und.edu
Dr. Ali Alshami
Assistant Professor
Harrington Hall Room 315
Phone: 701-777-6838
Email:

1
AN ANALYSIS OF THE IMPACT OF POLICY AND OTHER FACTORS ON
CARBON MONOXIDE EMISSIONS
Growing environmental concern throughout the world has ignited a number of
controversial views on how environmental cleaning should be achieved. In the context of
develop

Lecture 1-2
Summary Statistics
Navidi Chapter 1.2
1
Summary Statistics
Sample Mean:
n
1
X Xi
n i 1
Sample Variance:
n
n
2
1
1
2
2
2
s
Xi X
X i nX
n 1 i 1
n 1 i 1
Sample standard deviation is the square root of
the sample variance.
2
Group Problem
A

Lecture 4-2
Probability Plots
Navidi Section 4.5
1
Does your sample come from a
Normal Distribution?
If large number of samples, plot Histogram
Probability plots (discussed next)
If small number, difficult to determine
Samples from Normal distributions

Lecture 6-3
Hypothesis tests for differences between two
means
Navidi Chapter 6
1
Large Sample Tests for the Difference
Between Two Means
Now, we are interested in determining whether or not
the means of two populations are equal.
The data will consist

Lecture 1-3
Graphical Summaries
Navidi Chapter 1.3
1
Graphical Summaries
Stem-and-leaf plot
Dotplot
Histogram
Boxplot
Scatterplot
2
Stem-and-leaf Plot
A simple way to summarize a data set.
Each item in the sample is divided into two
parts: a stem, consi

Lecture 2-3:
Random Variables
Navidi Chapter 2.4
1
Random Variables
Definition: A random variable assigns a
numerical value to each outcome in a sample
space.
Definition: A random variable is discrete if its
possible values form a discrete set.
2
Example

Lecture 5-2
Small Sample Confidence Intervals
Navidi Chapter 5.3
1
Small Sample CIs for a Population
Mean
The methods that we have discussed for a population
mean previously require that the sample size be large.
When the sample size is small, there are

Lecture 2-2:
Counting Methods
Navidi Chapter 2.2
1
Counting Methods
The Fundamental Counting Principle:
Assume that k operations are to be
performed. If there are n1 ways to perform
the first operation, and if for each of these
ways there are n2 ways to p

Lecture 3-1:
Bias and Uncertainty
Navidi Chapter 3.1
1
Introduction
Any measuring procedure contains error.
This causes measured values to differ from the
true values that are measured.
Errors in measurement produce error in
calculated values (like the

Lecture 1-1:
Sampling
Navidi Chapter 1.1
1
Example 1
Example 1: Consider a machine that makes steel rods
for use in optical storage devices. The specification for
the diameter of the rods is 0.45 0.02 cm.
During the last hour, the machine has made 1000 r

Lecture 10-4
Mathematical Models from
Factorial Experiments
1
Equation (Model) for Factorial
b
Y
0
b1X1 b 2 X 2 b k X k
Constant term
Linear terms
b12X1X 2 b13X1X 3 b (k -1)k X k -1X k
Interaction terms
Where:
Xi is the coded value of Factor i, (-1, 0,

A
B
C
D
250
263
257
253
264
254
279
258
256
267
269
262
260
265
273
264
239
267
277
273
A
B
C
D
250
263
257
253
264
254
279
258
I
J
N
256
267
269
262
260
265
273
264
239
267
277
273
4
5
20
df
SS
Treatment
Error
Total
3
16
19
MS
743.4
1023.6
1767
F
p
F*
24

Lecture 10-3
Factorial Experiments
1
Typical Steps in an Experimental Design Process
1.
2.
3.
4. (sometimes)
Why
Why start
start in
in the
the middle?
middle?
Screening
Designs
Fr
ac
ti
on
ate
1.
2.
2-Level Factorial
Designs
t
en
gm
Au
3.
Optimization
Des