Download Populations and Samples Chapter 8

Fundamental Sampling Distributions
 Introduction to random sampling and statistical
inference
 Populations and samples
 Sampling distribution of means
 Central Limit Theorem
 Other distributions
 S2
 t-distribution
 F-distribution
EGR 252 - Ch. 8
9th edition
Spring 2013
Slide 1
Populations and Samples
 Population: “a group of individual persons, objects, or
items from which samples are taken for statistical
measurement” 1
 Sample: “a finite part of a statistical population whose
properties are studied to gain information about the
whole” 1
 Population – the totality of the observations with which
we are concerned 2
 Sample – a subset of the population 2
1 (Merriam-Webster Online Dictionary, http://www.m-w.com/, October 5, 2004)
2 Walpole, Myers, Myers, and Ye (2007) Probability and Statistics for Engineers and Scientists
EGR 252 - Ch. 8
9th edition
Spring 2013
Slide 2
Examples
Population
Students pursuing
undergraduate
engineering degrees
Sample
1000 engineering
students selected at
random from all
engineering programs in
the US
Cars capable of speeds in 50 cars selected at
excess of 160 mph.
random from among
those certified as having
achieved 160 mph or
more during 2003
EGR 252 - Ch. 8
9th edition
Spring 2013
Slide 3
More Examples
Population
Sample
Potato chips produced at
the Frito-Lay plant in
Kathleen
10 chips selected at
random every 5 minutes
as the conveyor passes
the inspector
Freshwater lakes and
rivers
4 samples taken from
randomly selected
locations in randomly
selected and
representative freshwater
lakes and rivers
EGR 252 - Ch. 8
9th edition
Spring 2013
Slide 4
Basic Statistics (review)
n
Sample Mean:
X
X
i 1
i
n
A class project involved the formation of three 10-person teams (Team
Q, Team R and Team S). At the end of the project, team members
were asked to give themselves and each other a grade on their
contribution to the group. A random sample from two of the teams
yielded the following results:
Q
R
X Q = 87.5
92 85
95
88
85
75
78
92
EGR 252 - Ch. 8
9th edition
XR
= 85.0
Spring 2013
Slide 5
Basic Statistics (review)
 Sample variance equation:
n
S2 
 For our example:
2
(
X

X
)
 i
i 1
n 1
Q team sample R team sample
92
85
95
88
85
75
78
92
 Calculate the sample standard deviation (s) for each sample.
 SQteam = 7.59386 and SRteam = 7.25718
EGR 252 - Ch. 8
9th edition
Spring 2013
Slide 6
Sampling Distributions
If we conduct the same experiment several
times with the same sample size, the probability
distribution of the resulting statistic is called a
sampling distribution
Sampling distribution of the mean: if n
observations are taken from a normal population
with mean μ and variance σ2, then:
      ...  
x 

n
2
2
2
2
2
      ...

2
x 

2
n
n
EGR 252 - Ch. 8
9th edition
Spring 2013
Slide 7
Central Limit Theorem
Given:
X : the mean of a random sample of size n taken from a
population with mean μ and finite variance σ2, the
limiting form of the distribution of
X 
Z 
, n  
/ n
is the standard normal distribution n(z;0,1)
Note that this equation for Z applies when we have
sample data.
Compare to the Z equation for the population (Ch6).
EGR 252 - Ch. 8
9th edition
Spring 2013
Slide 8
Central Limit Theorem-Distribution of X
If the population is known to be normal, the
sampling distribution of X will follow a normal
distribution.
Even when the distribution of the population
is not normal, the sampling distribution of X
is normal when n is large.
 NOTE: when n is not large, we cannot assume the
distribution of X is normal.
EGR 252 - Ch. 8
9th edition
Spring 2013
Slide 9
Sampling Distribution of the Difference
Between Two Averages
Given:
 Two samples of size n1 and n2 are taken from two
populations with means μ1 and μ2 and variances σ12
and σ22
Then,
 X  X  1   2
1
 X2
2
1X 2

 12
n1

 22
n2
and
Z
( X 1  X 2 )  ( 1   2 )
 12
n1
EGR 252 - Ch. 8
9th edition

 22
n2
Spring 2013
Slide 10
Sampling Distribution of S2
Given:
 If S2 is the variance of of a random sample of size n
taken from a population with mean μ and finite
variance σ2,
Then,
2 
(n  1) s 2

2
n

i 1
(Xi  X )
2
2
has a χ 2 distribution with ν = n - 1
EGR 252 - Ch. 8
9th edition
Spring 2013
Slide 11
Chi-squared (χ2) Distribution
α
χ2
 χα2 represents the χ2 value above which we find an area
of α, that is, for which P(χ2 > χα2 ) = α.
EGR 252 - Ch. 8
9th edition
Spring 2013
Slide 12
Example
 Look at example 8.7, pg. 245: A manufacturer of car batteries
guarantees that his batteries will last, on average, 3 years with a
standard deviation of 1 year. A sample of five of the batteries yielded
a sample variance of 0.815. Does the manufacturer have reason to
suspect the standard deviation is no longer 1 year?
μ = 3 σ = 1 n = 5 Degrees of freedom (v) = n-1
s2 = 0.815
calculated  
2
(n  1) s 2
2

(4)(0.815)
 3.26
1
If the χ2 value fits within an interval that covers 95% of the χ2 values
with 4 degrees of freedom, then the estimate for σ is reasonable.
See Table A.5, (pp. 739-740)
For alpha = 0.025, Χ2 =11.143
The Χ2 value for alpha = 0.975 is 0.484.
EGR 252 - Ch. 8
9th edition
χ2
0.484 3.26
Spring 2013
11.143
Slide 13
Your turn …
If a sample of size 7 is taken from a normal
population (i.e., n = 7), what value of χ2
corresponds to P(χ2 < χα2) = 0.95? (Hint: first
determine α.)
95%
χ2
12.592
EGR 252 - Ch. 8
9th edition
Spring 2013
Slide 14
t- Distribution
 Recall, by Central Limit Theorem:
X 
Z
/ n
is n(z; 0,1)
 Assumption: _____________________
(Generally, if an engineer is concerned with a familiar
process or system, this is reasonable, but …)
EGR 252 - Ch. 8
9th edition
Spring 2013
Slide 15
What if we don’t know σ?
New statistic:
X 
T
S/ n
Where,
n
X 
i 1
Xi
n
( Xi  X )
and S  
n 1
i 1
n
2
follows a t-distribution with ν = n – 1 degrees of
freedom.
EGR 252 - Ch. 8
9th edition
Spring 2013
Slide 16
Characteristics of the t-Distribution
Look at Figure 8.8, pg. 248
Note:
 Shape: _________________________
 Effect of ν: __________________________
See table A.4, pp. 737-738 Note that the table
yields the right tail of the distribution.
EGR 252 - Ch. 8
9th edition
Spring 2013
Slide 17
F-Distribution
Given:
 S12 and S22, the variances of independent random
samples of size n1 and n2 taken from normal
populations with variances σ12 and σ22, respectively,
Then,
S12 /  12  22S12
F 2 2  2 2
S2 /  2  1 S2
has an F-distribution with ν1 = n1 - 1 and ν2 = n2 – 1
degrees of freedom.
(See table A.6, pp. 741-744)
EGR 252 - Ch. 8
9th edition
Spring 2013
Slide 18