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
Continuous Distributions
1
Normal Distribution
2
Normal Distribution
Reasonable approximation for many reallife, continuous experimental outcomes.
heights, weights
measurement errors
reaction times
3
Normal Distribution
1
e (1 / 2)[( x m ) / s]
n(x; m, s
Statistical Methods for Product and Processes Development
ENME 392

Spring 2013
ENME392 Spring 2013
Homework 3
Due Tuesday February 12 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
Statistical Methods for Product and Processes Development
ENME 392

Spring 2013
ENME392 Spring 2013
Homework 4
Due Tuesday February 19 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
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
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
Statistical Methods for Product and Processes Development
ENME 392

Spring 2013
ENME392 Spring 2013
Homework 2
Due Tuesday February 5 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 soluti
ENME392 Fall 2011
Homework 13
Due Tuesday December 6th at the beginning of the 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
t
ENME392 Fall 2011
Homework 3
Due Tuesday September 20th at the beginning of the 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
Coin Toss Data
1
Coin Toss Data
21 groups tossed 10 coins 9 times (9 samples were taken) and we recorded
the average number of heads for each group.
Group
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Average #
Heads
5.333
4.444
4.333
5.000
5.111
Sampling
1
Sampling Distribution of Means
2
Central Limit Theorem
If X is the mean of a random sample of size n
taken from a population with mean m and variance s2, then
the limiting form of the distribution of Z as n is n(z;0,1),
the standard normal dist
Estimation
1
Estimation
Up to now:
probability
population
sample
statistical inference
2
Estimation
Now: what is the true value, based on the sample data?
Example: The outcome of an election, based on polling data.
probability
population
sample
statistica
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
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