Parameter Estimation
Point Estimation
Let X 1 ,L, X n be a random sample from some
population, characterized by the unknown
parameter .
Let = h( X 1 ,L, X n ) be some f n of the sample data.
is a statistic which is an estimator of .
Example:
X i ~ iid N
Homework on Simple Linear Regression
A researcher reported the median grain size of sand (in mm) in 59 alluvial
aquifers in the Arkansas River Valley. The yield of each aquifer (in gal/day/ft2)
was also reported. The data are contained in the file titled
Homework on 2 2 Table
The voting results for the U. S. Senate confirmation of Clarence Thomas as a
justice of the Supreme Court are summarized below in a 2 2 contingency table
showing the voting results by political party affiliation. Test the hypothesis
HW-4
a. The sample does not appear to be normal, and using the rough statistic that if n30,
assume normality due to the central limit theorem, does not apply. In additoin a shipiro
wilkes test returns a p-value of .1452, which indicates a non-normal distr
HW-7
1.
Stat 537 Page 1
2.
1 = 744.98
0 = -9.294
3.
As seen above, the 95% confidence interval is shaded, and p <0.001 so we will reject the null hypothesis
and indicate that 1 0 with 95% conficence with a t-statistic = 6.77 (or (F value)
4.
Based on the
HW-4&5
a. The sample does not appear to be normal, and using the rough statistic that if n30,
assume normality due to the central limit theorem, does not apply. In additoin a shipiro
wilkes test returns a p-value of .1452, which indicates a non-normal dis
Homework on Correlation Coefficients
A researcher reported the median grain size of sand (in mm) in 59 alluvial
aquifers in the Arkansas River Valley. The yield of each aquifer (in gal/day/ft2)
was also reported. The data are contained in the file titled
Introduction to Statistics
2
1.
Fundamental Statistical Concepts
Objectives
Decide what tasks to complete before analyzing
the data.
Use the Summary Statistics task to produce
descriptive statistics.
3
Defining the Problem
The purpose of the study is to d
Discrete Random Variables
Def n of a discrete random variable
Consider an experiment having a finite (or countably
infinite) number of possible outcomes.
Assign some real number to each possible outcome,
and associate this number with some variable, say X
Sample Statistics
Let X 1 , X 2 , ", X n be a random sample of size n from
some population with probability distribution f(x; ).
Let X (1) X (2) " X ( n ) denote the order statistics from
this sample.
Mean
1 n
1
x = xi = ( x1 + x2 + " + xn )
n i =1
n
Vari
Correlation
The correlation between two random variables X and Y is
defined as
XY
xy
Cov( X , Y )
=
=
Var ( X ) Var (Y ) x y
where
cfw_
Cov( X , Y ) = xy = E [ x x ] y y
cfw_
Var ( X ) = 2x = E [ x x ]
2
Properties of
(1) 1 1
(2) is independent of t
Hypothesis Testing
Hypothesis - A statement concerning one or more
populations
Statistical Hypothesis - consists of a null hypothesis and
an alternative hypothesis
Null hypothesis - denoted by H0
states claim to be tested and is often
formulated with hope
Alternative Nonparametric Tests
One Sample Test of Location Sign Test
x1 , x2 ,
xn ~ iid Fx (continuous)
The sign test tests an hypothesis about pop n median :
H0 : = 0
H1 : 0
Equivalent to
H 0 : P [ x < 0 ] = 0.5
Use binomial test
H 0 : = 0.5
with test s
Categorical Data Analysis
C.I. on the population proportion
We assume that
(1 )
~ N ,
n
true for large n
So that
P z 2
z 2 = 1
(1 )
n
We need to solve
= z 2
(1 )
n
for ; which yields the following quadratic equation
in :
(1 )
n
z2 2 2
z2 2
2
Continuous Random Variables
Def n of a continuous random variable
Consider an experiment having an infinite number of
possible outcomes.
Assign some real number to each possible outcome,
and associate this number with some random variable,
say X.
Example:
HW-6
10.12 An entomology PhD student is studying rare spider species. She would like to determine the
population density of the spider in a particular region in South Dakota. She sets out 20 traps in randomly
selected locations within the specified region