Fundamental Sampling Distributions
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Transcript of Fundamental Sampling Distributions
EGR 252 - Ch. 8 9th edition 2013 Slide 1
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 2013 Slide 2
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 2013 Slide 3
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 excess of 160 mph.
50 cars selected at random from among those certified as having achieved 160 mph or more during 2003
EGR 252 - Ch. 8 9th edition 2013 Slide 4
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 2013 Slide 5
Basic Statistics (review)
Sample Mean:
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:
= 87.5
= 85.0
Q R
92 85
95 88
85 75
78 92
n
XX
n
ii
1
QX
RX
EGR 252 - Ch. 8 9th edition 2013 Slide 6
Basic Statistics (review) Sample variance equation:
For our example:
Calculate the sample standard deviation (s) for each sample.
SQteam = 7.59386 and SRteam = 7.25718
Q team sample R team sample
92 85
95 88
85 75
78 92
1
)(1
2
2
n
XXS
n
ii
EGR 252 - Ch. 8 9th edition 2013 Slide 7
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:
nn
n
x
x
2
2
22222 ...
...
EGR 252 - Ch. 8 9th edition 2013 Slide 8
Central Limit TheoremGiven:
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
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).
nn
XZ ,
/
EGR 252 - Ch. 8 9th edition 2013 Slide 9
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 2013 Slide 10
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,
2
22
1
21
2121
2
22
1
212
21
)()(
21
21
nn
XXZ
and
nnXX
XX
EGR 252 - Ch. 8 9th edition 2013 Slide 11
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,
has a χ 2 distribution with ν = n - 1
n
i
i XXsn
12
2
2
22 )()1(
EGR 252 - Ch. 8 9th edition 2013 Slide 12
Chi-squared (χ2) Distribution
χα2 represents the χ2 value above which we find an area
of α, that is, for which P(χ2 > χα2 ) = α.
χ2
α
EGR 252 - Ch. 8 9th edition 2013 Slide 13
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
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)χ2
26.31
)815.0)(4()1(2
22
sn
calculated
For alpha = 0.025, Χ2 =11.143 The Χ2 value for alpha = 0.975 is 0.484. 0.484 3.26 11.143
EGR 252 - Ch. 8 9th edition 2013 Slide 14
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 < χα
2) = 0.95? (Hint: first determine α.)
χ2
12.592
95%
EGR 252 - Ch. 8 9th edition 2013 Slide 15
t- Distribution
Recall, by Central Limit Theorem:
is n(z; 0,1)
Assumption: _____________________
(Generally, if an engineer is concerned with a familiar process or system, this is reasonable, but …)
n
XZ
/
EGR 252 - Ch. 8 9th edition 2013 Slide 16
What if we don’t know σ?
New statistic:
Where,
and
follows a t-distribution with ν = n – 1 degrees of freedom.
nS
XT
/
n
i
i
n
XX
1
n
i
i
n
XXS
1
2
1
)(
EGR 252 - Ch. 8 9th edition 2013 Slide 17
Characteristics of the t-Distribution
Look at Figure 8.8, pg. 248Note:
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 2013 Slide 18
F-DistributionGiven:
S12 and S2
2, the variances of independent random samples of size n1 and n2 taken from normal populations with variances σ1
2 and σ22, respectively,
Then,
has an F-distribution with ν1 = n1 - 1 and ν2 = n2 – 1 degrees of freedom.
(See table A.6, pp. 741-744)
22
21
21
22
22
22
21
21
/
/
S
S
S
SF