The Chi-Square Distribution
The Chi-Square Distribution
1
If Z N(0, 1), then Z 2 21 .
The Chi-Square Distribution
1
2
If Z N(0, 1), then Z 2 21 .
If Z1 , Z2 , ., Zk are independent N(0, 1) random variables,
then X = Z12 + Z22 + . + Zk2 has a 2k distributi
Statistics 100B
Midterm Study Guide
For the midterm, you should know.
1. The Basics
How to use proper notation
How to interpret the text of a problem into proper notation
How to use mathematical operators properly, e.g. equal signs should only be between
Two Important Results from Normal Theory
Suppose a random sample of size n is taken from a normal
population with mean , and standard deviation . Then,
Two Important Results from Normal Theory
Suppose a random sample of size n is taken from a normal
popul
Homework Solution
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Estimating E (y ) and y from x
Estimating E (y ) and y from x
Suppose we have constructed the regression model y = a + bx
for some bivariate data, and that we are satisfied with our
assumptions.
Estimating E (y ) and y from x
Suppose we have constructed t
Randomized Block Design (RBD)
1
Recall our polymer example for the Completely Randomized
Design.
Randomized Block Design (RBD)
1
Recall our polymer example for the Completely Randomized
Design.
2
In this example, we assumed that the experimental units
(EU
The Wilcoxon Rank Sum Test
The Wilcoxon Rank Sum Test
Suppose we want to compare the distributions of two
populations.
The Wilcoxon Rank Sum Test
Suppose we want to compare the distributions of two
populations.
In particular, we are interested in whether
Statistics 100B
Final Exam Study Guide
For the Final Exam, you should know.
1. The Basics
a. How to use proper notation
b. How to interpret the text of a problem into proper notation
c. How to use mathematical operators properly
d. Basic Summation Notatio
What is a Distribution Function?
What is a Distribution Function?
1
Every (real-valued) random variable has a Distribution
Function (or DF for short), uniquely defining it.
What is a Distribution Function?
1
Every (real-valued) random variable has a Distr
The Sign Test
Suppose we are presented with the following data, where some
treatment is applied to each subject between Before and After.:
Subject
Tim
Jen
Ben
Ed
Hal
Ann
Lee
Les
Rex
Before
13
15
9
16
19
13
18
19
14
After
12
14
16
16
18
11
16
16
13
Subject
September 25, 2015 Slides
The Uniform Distribution
Let U U(0, 1), i.e. U has density f (x ) = 1, for 0 < x < 1, and 0
otherwise:
What is P(U < 0.5)?
The Uniform Distribution
P(U < 0.5) = 0.5
What is P(U < 0.328)?
The Uniform Distribution
P(U < 0.328) = 0.
Statistics 100B - Midterm Formula Sheet
P
x=
n
x 0
z=
z=
s
n
x1 x2 D0
q 2
s22
s1
n1 + n2
t=
F =
(n1 1)s21 + (n2 1)s22
=
n1 + n2 2
n 1s2
02
SST =
k
X
s
x1 x2 t 2 ,[min(n1 ,n2 )1]
x1 x2 t 2 ,[n1 +n2 2] sp
d t 2 ,n1
sd
n
s21
s2
+ 2
n1
n2
r
d d
s
d
n
i
h
c.v.
Factorial Design
1
Once again, recall our polymer example.
Factorial Design
1
Once again, recall our polymer example.
2
We return to assuming that the experimental units (EUs)
have no major differences, i.e. we need not block.
Factorial Design
1
Once agai
Categorical Data Analysis
Categorical Data Analysis
Is superstitious behavior uniquely human?
Categorical Data Analysis
Is superstitious behavior uniquely human?
In the 1940s, a Harvard psychiatrist set out to determine if
another creature, a pigeon, can
ANOVA for Hubbles Data
Source
DF
Sum Sq.
Mean Sq.
F
p-Value
Regression
1
1976648
1976648
36.438
p < 0.005
Error
22
1193442
54247
Total
23
3170090
Test for = 0
Suppose we wish to test if there is a significant linear relationship
between distance and veloc
ANOVA for Simple Linear Regression
1
Consider Again Hubbles Data.
ANOVA for Simple Linear Regression
1
Consider Again Hubbles Data.
2
Recall that we found the least-squares linear model for this
data to be y = 40.78365 + 454.158x .
ANOVA for Simple Linear
Point Estimation of Population Parameters
Point Estimation of Population Parameters
1
Suppose X is a random variable with unknown mean and
unknown standard deviation .
Point Estimation of Population Parameters
1
Suppose X is a random variable with unknown
Example
Suppose an engineer wishes to test whether an additive affects
hardness of a particular polymer. In order to do this, the engineer
prepares ten solutions of polymer, randomly selects five, and
introduces the additive in them. She then measures the
RE: Homework 2A
Two Choices:
RE: Homework 2A
Two Choices:
Turn it in now for a maximum of 40 points out of 30.
RE: Homework 2A
Two Choices:
Turn it in now for a maximum of 40 points out of 30.
Continue working on it until Wednesday for a maximum of 30
poi
STAT 100B
Chapter 12: Linear Regression and Correlation
Example: An experiment was conducted to examine the
effect of different concentrations of pectin on the firmness
of canned sweet potatoes. Three concentrations were
used (0%, 1.5%, and 3%). Six numb
STAT 100B
Chapter 12: Linear Regression and Correlation
In Chapter 11, we used ANOVA to investigate the
effect of various treatments (factor levels or factor
level combinations) on a response variable.
ANOVA tells us if there is a relationship between t
STAT 100B
Section 11.9: The a x b Factorial Experiment: A Two-Way Classification
In a two-way factorial design, two factors of interest, A
and B, are compared at several levels. Each treatment
(factor-level combination) is replicated r times to allow
for
STAT 100B
Section 11.7-11.8: Randomized Block Design
In a randomized block design, a two-way analysis of
variance (ANOVA) is used to compare k treatments
within b relatively homogenous groups of experimental
units called blocks.
We still need the treatme
STAT 100B
Section 11.6: Ranking Population Means
Many experiments are exploratory in nature. You have no
preconceived notions about the results and have not
decided (before conducting the experiment) to make
specific treatment comparisons.
If the main AN
STAT 100B
Chapter 11: The Analysis of Variance
11.1: The Design of an Experiment
From 7.2: Sampling Plans & Experimental Designs
observational
study - the researcher
does not actually produce the data but
only observes the characteristics of data
that a
STAT 100B
Section 10.5: Small-Sample Inferences for the Difference between Two Population
Means: A Paired-Difference Test
Samples are dependent or paired when
the observations are collected in pairs
or
the observations in one sample are naturally
relate