Discrete Distributions
1
Binomial Distribution
2
Binomial Distribution: Mean and Variance
b(x; n,p) =
n!
pxqn x
.
x!(n x )!
where q = 1p
m = E(X) = np
s2 = E(X2) m2 = npq
Example for m
If you toss 10 coins, how many heads to you expect, on average?
10 co
Statistical Methods for Product and Processes Development
ENME 392

Spring 2011
ENME392 Spring 2011
Homework 6 Solutions
For this problem, you may work in groups of up to 4. Obtain a box or box lid of
approximately 9 x 13. Draw a square on the bottom of it that is 3.5 on each side; the
square can be off center. From a height of appro
Estimation
1
Paired Observations
2
Paired Observations
The experiment is designed to look at differences between paired observations.
before/after studies on same individuals
two kinds of tire on one car
studies using twins
This reduces the variance du
Statistical Methods for Product and Processes Development
ENME 392

Spring 2013
ENME392 Spring 2013
Homework 1
Due Tuesday January 29 in class
Notes:
1. You may not look at solutions from previous years, since many of the problems assigned
here have been used before. The honor code applies, even if you do not need to include it
on yo
ENME392 Fall 2013
Homework 1 Solutions
Assignment:
1) (25 pts) Using the dataset of exercise 1.1 in Walpole:
3.4, 2.5, 4.8, 2.9, 3.6, 2.8, 3.3, 5.6, 3.7, 2.8, 4.4, 4.0, 5.2, 3.0, 4.8
a) (3 pts) What is the sample size?
Sample size (n) = 15
b) (3 pts) Calc
ENME392 Fall 2013
Homework #3 Solutions
1. Chapter 3: 3.39 (5 pts)
sack of fruit with 3 oranges X, 2 apples Y, 3 bananas B
8 total fruit, take 4 pieces
X = # oranges, Y = # of apples
(a) f(x,y) = ?
same as example 3.14 p. 92
3 2 3
x y 4xy for x = 0
ENME392 Fall 2013
Homework 5
Due Tuesday October 8th in class
Notes:
1. You may not look at solutions from previous years, since many of the problems assigned
here have been used before. The honor code applies and you must include the honor
code pledge (p
ENME392 Fall 2013
Homework 6 Solutions
For this assignment, you may work in groups of up to 4. Obtain a small cardboard box (with
sides intact) or box lid of approximately 9 x 13. Draw a square on the bottom that is 4.0 on
each side; the square can be off
ENMA 300/ENME 382
Homework 6
Due: Monday Oct 28, 2013
* Read Chapter 6, Callister 4th edition.
DIFFUSION
1a. For substitutional diffusion of zinc in copper, Do = 2.4 x 105 cm2/s and Ea = 1.96 eV. Calculate the
diffusion coefficient of zinc in copper at 6
ENME 392, Fall 2014
Statistical Methods for Product and Processes Development
Instructor: Professor Peter W. Chung
Youve heard it said
There are three kinds of lies: lies, damned lies, and statistics.*
Why is statistics a required course?
What do engineer
Probability
Understanding the basic principles behind
calculating and working with probabilities
1
Definitions
Events are sets of outcomes
Experiment is a process that generates data
Random Experiment is an experiment that, when repeated in the same
ma
Continuous Random Variables and
Continuous Probability Distributions
Understand how continuous probability distributions
are developed and how they are used/analyzed
1
Recap(itulation)
How to count
Sample spaces, Permutations, Combinations, Addition & M
Discrete Random Variables and
Discrete Probability Distributions
Understand how discrete probability distributions are
developed and how they are used/analyzed
1
Recap(itulation)
How to count
Sample spaces, Permutations, Combinations, Addition & Multipl
Linear Regression Using Excel
Example
All of this can also be done automatically in Excel.
Go to Tools
Data Analysis
Regression
Choose the y data, then the x data.
Example
X Variable 1 Line Fit Plot
SUMMARY OUTPUT
350
Regression Statistics
Multiple R
0.98
Continuous Random Variables and
Continuous Probability Distributions
Understand how continuous probability distributions
are developed and how they are used/analyzed
1
Recap(itulation)
How to count
How to determine single point probabilities
Sample spaces
Probability
Distributions
Specific Distributions
Specific distributions
Binomial
Multinomial
Hypergeometric
Poisson
Binomial Distribution
Binomial Distribution
Describes
events arising from series of
random experiments called the Bernoulli
Process.
Statistical Methods for Product and Processes Development
ENME 392

Spring 2013
ENME392 Spring 2013
Homework 5
Due Tuesday February 26 in class
Notes:
1. You may not look at solutions from previous years, since many of the problems assigned
here have been used before. The honor code applies.
2. Please show all your steps in the solut
Estimation
1
Estimating Variances
2
Estimating the Variance
2
( n 1 )s 2
s2
If s2 is the variance of a random sample of size n from a normal population, a
100(1a)% confidence interval for s2 is
v = n  1 d.o.f.
( n 1 )s 2
a / 2
2
s
2
( n 1 )s 2
12a / 2
Hypothesis Testing
1
Topics for Todays Lecture
1. Type I and II Errors
2. Tests on one mean
3. Tests on two means
2
Type I and II Errors
Type I error
Rejection of H0 when it is true.
Probability of committing = = level of significance
Type II error
Accept
Hypothesis Testing
1
2
Two Approaches
One sets based on how sure one wants to be about rejecting H0.
Typical values are 0.05 and 0.01
Instead of just accepting or rejecting H0, one can provide the probability of
rejecting H0, when H0 is true
Fixed pproach
Regression Analysis
Simple Linear Regression
Method of Least Squares
Quality of Fit
Multiple Linear Regression
Excel for Data Analysis
1
Regression Analysis
Regression analysis is used to establish a relationship between quantities or
populations. You
ANOVA
1
2
ChiSquared vs. Analysis of Variance
Analysis of variance (ANOVA) is a collection of statistical models, and their
associated procedures, in which the observed variance in a particular variable
is partitioned into components attributable to diff
ANOVA
1
Review:
1Way ANOVA Model
Test the null hypothesis:
Test statistic:
s12
f= 2,
s
H0: 1 = 2 = 3 = . = k = 0
H1: at least one of the k 0
SSA
s12 =
where
k 1
s2 =
SSE
k( n 1 )
If H0 is true, then s12 s2 2
But if H1 is true, then s12 > s2
Reject H0 at
ANOVA
1
Review:
1. 1Way ANOVA Model
Test the null hypothesis:
Test statistic:
s12
f= 2
s,
H0: 1 = 2 = 3 = . = k
H1: at least two of them are not equal
SSA
s12 =
where
k 1
s2 =
SSE
k( n 1 )
If H0 is true, then s12 s2 2
But if H1 is true, then s12 > s2
Rej
ANOVA
Block Designs
1
Randomized Complete Block Designs (RCB)
What if there is another source of variation, but it is one in which we are not
interested? We put these variations into blocks.
Blocking is used to control for sources of variation such as
di
Design of Experiments
1
Purpose of DOE
Screening a large number of effects (or variables) to see which to investigate
further. How to do this in the most efficient way?
DOE is used to:
obtain unambiguous results at minimum cost
learn about interactions
ENME392 Fall 2011
Homework 1
Due Tuesday September 6th in class
Notes:
1. You may not look at solutions from previous years, since many of the problems assigned
here have been used before. The honor code applies and you must include the honor
code pledge
Math 221 Calculus III Summer 2016 Final Examination (Take home portion)
/ 250
Do all work on separate paper with you r name on EACH page; attach to test. All answers must
be justified. Lack of work shown may mean loss of cred i t. Poi nt val ues are indic
ENME 392 Fall 2016
Written Homework #4 Confidence Intervals
DUE: Tuesday October 4, 1159PM
Instructions:
Written assignments will be collected and graded ELECTRONICALLY via Canvas.
RULES:
Homework must be uploaded by 1159pm on the day it is due.
Illegib
Exam 2 Question #19 required as part of the exam. NAME_
19. The base of an aquarium with a given volume V is made of slate and the sides are made of glass.
If slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium tha
ENME 392 Fall 2016
Written Homework #6 Hypothesis Testing II
DUE: Thurs October 27, 1159PM
Instructions:
Written assignments will be collected and graded ELECTRONICALLY via Canvas.
RULES:
Homework must be uploaded by 1159pm on the day it is due.
Illegib
ENME 392
Fall 2016
Written Homework #9 DOE
DUE: Tuesday Dec 6, 11:59PM
Instructions:
Written assignments will be collected and graded ELECTRONICALLY via Canvas.
RULES:
Homework must be uploaded by 1159pm on the day it is due.
Illegible scans will NOT be