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
 Data displays / Graphical methods
EGR 252 - Ch. 8 Part 1 and 2
Spring 2009
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 Part 1 and 2
Spring 2009
Slide 2
Examples
Population
Sample
Students pursuing
undergraduate
engineering degrees
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 Part 1 and 2
Spring 2009
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 Part 1 and 2
Spring 2009
Slide 4
Basic Statistics (review)
n
Sample Mean:
X
X
i 1
i
n
 At the end of a team project, team members were asked
to give themselves and each other a grade on their
contribution to the group. The results for two team
members were as follows:
Q
S
92
85
95
88
85
75
78
92
EGR 252 - Ch. 8 Part 1 and 2
X Q = ___________________ 87.5
X S = ___________________ 85.0
Spring 2009
Slide 5
Basic Statistics (review)
1. Sample Variance:
n
S2 
( X
i 1
i
 X )2
n 1

n
n
i 1
i 1
n  X i2  ( X i )2
n(n  1)
 For our example:
Q
S
92
85
95
88
85
75
78
92
SQ2 = ___________________
SS2 = ___________________
 S2Q = 7.59386
 S2S = 7.25718
EGR 252 - Ch. 8 Part 1 and 2
Spring 2009
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 Part 1 and 2
Spring 2009
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,
Then,
 the limiting form of the distribution of
X 
Z 
, n  
/ n
is the standard normal distribution n(z;0,1)
EGR 252 - Ch. 8 Part 1 and 2
Spring 2009
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 Part 1 and 2
Spring 2009
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
2
1 X 2


n1
2
2
n2
and
Z
( X 1  X 2 )  ( 1  2 )
1
2
n1
EGR 252 - Ch. 8 Part 1 and 2

2
2
n2
Spring 2009
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 Part 1 and 2
Spring 2009
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 Part 1 and 2
Spring 2009
Slide 12
Example
 Look at example 8.10, pg. 256: 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 s2 = 0.815
 
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. 755-756)
Χ20.025 =11.143
EGR 252 - Ch. 8 Part 1 and 2
χ2
Χ20.975 = 0.484
Spring 2009
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.)
χ2
12.592
EGR 252 - Ch. 8 Part 1 and 2
Spring 2009
Slide 14
t- Distribution
 Recall, by CLT:
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 Part 1 and 2
Spring 2009
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 Part 1 and 2
Spring 2009
Slide 16
Characteristics of the t-Distribution
Look at fig. 8.11, pg. 221***
Note:
 Shape: _________________________
 Effect of ν: __________________________
See table A.4, pp. 753-754
EGR 252 - Ch. 8 Part 1 and 2
Spring 2009
Slide 17
Comparing Variances of 2 Samples
Given two samples of size n1 and n2, with
sample means X1 and X2, and variances, s12 and
s 22 …
Are the differences we see in the means due to
the means or due to the variances (that is, are
the differences due to real differences between
the samples or variability within each samples)?
See figure 8.16, pg. 226
EGR 252 - Ch. 8 Part 1 and 2
Spring 2009
Slide 18
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. 757-760)
EGR 252 - Ch. 8 Part 1 and 2
Spring 2009
Slide 19
Data Displays/Graphical Methods
Box and Whisker Plot
 Page 236
 Min Max values
 Q1 Q2 Q3
 Interquartile range
Quantile-Quantile Plot
Normal Probability Plot
Minitab
EGR 252 - Ch. 8 Part 1 and 2
Spring 2009
Slide 20