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Math 4030 – 8a
Population and Sample
Sample Mean Distribution
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Sample in terms of iid RV (Sec. 6.1):
Population: a random variable X with certain
distribution (discrete or continuous);
Sample (of size n): n independent random variables
that have the same (identical) distribution as X.
A random sample of size n can be viewed as an
n-dimensional random vector of which all
components have the independent and identical
distribution, called population or underlying
distribution.
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Parameter vs. Statistic
• Parameters are numbers that
summarize data for an entire
population.
• Statistics are numbers that
summarize data from a sample, i.e.
some subset of the entire
population.
Sample statistics are random
variables, while population
parameters are not.
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An example of sample means:
Population: S = {0, 1, 2, 3, 3, 5, 6, 9}
Consider random samples of size 3, the following
samples are equal likely to be formed:
{012, 123, 013, 015, 016, 019,
023, 123, 025, 026, 029,
033, 035, 036, 039,
035, 036, 039,
056, 059,
069;
233, 235, 236, 239,
235, 236, 239,
256, 259,
269;
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123, 123, 125, 126, 129,
133, 135, 136, 139,
135, 136, 139,
156, 159,
169;
335, 336, 339,
356, 359,
369;
356, 359,
369;
569}
 8
876
 56
 8 C3 
3  2 1
 3
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Sampling distribution of the Mean (Sec. 6.2)
X has any distribution with the mean µ and
standard deviation . Let
n
1
X   Xi
n i 1
be the sample mean from an (independent)
sample of size n. Then
 X  E  X   
and  2 X
2

 Var  X  
.
n
If the population is finite, sample
variables cannot be independent.
However we still have,
 X  E  X   

and

N n

 Var  X  


2
2
X
 
n  N 1 
Finite Population
Correction Factor
Chebyshev’s Theorem:
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P  X   X  k X   2 , for all k  1.
k
X  Xn,
X  X ,  X 
n
n
X
n
,
X  1

P X n   X  k
  2 , for all k  1.
n k

 k

X

n


 X 
P X n  X    

 n 
2
lim n P X n   X    0, for any   0.
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Law of Large Number (Theorem 6.2):
X1,X2,…, Xn is a sample from a population with
(finite) mean  and (finite) variance 2, then for
any arbitrary (small and) positive number ,


lim P X      0,
n 
where
1 n
X   Xi
n i 1
is the sample mean.
(Long-run) relative frequency and probability.
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The Central Limit Theorem (Theorem 6.3)
X has any distribution with the mean µ and standard
deviation . Let
1 n
X   Xi
n i 1
be the sample mean from an (independent) samples
of size n. Then
 2 
X 
X  N   ,  or Z 
 N 0,1
n 
/ n

if n is large.
Further more, if the population variance is unknown, we
may use the sample standard deviation. i.e.
X 
Z
 N 0,1
s/ n
if n is large. (n ≥ 30)
Sample mean distribution
If the population is
normally
distributed with
known mean and
variance, then the
sample mean is
normally
distributed.
For large sample
(at least 30), the
sample mean is
approximately
normally
distributed
(Central Limit
Theorem).
If population is normally distributed with
unknown variance,
X 
t
 t n  1
S/ n
Where S is the sample standard
deviation, and t(n-1) is the tdistribution with degree of freedom
n-1.
(Sec. 6.3)
X ~ t ( )
Excel:
P(X < t) :
t:
R:
P(X < t) :
t:
Table:
T.DIST(t, , T)
T.INV(1-,)
pt(t,)
qt(1-,)
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Distribution of Sample Means:
 2 
X 

Z
 N 0,1 or X  N   ,
n 
/ n

Use Table 3
X 
t
 t n  1.
s/ n
Use Table 4 for n < 30
and Table 3 for n ≥ 30.
Population
Identify the population
according to our
research objective(s).
Based on the data analysis
results, make inference
toward the population (e.g.
estimation/prediction)
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Draw a sample: to
ensure the sample
preserves the
same
characteristics as
that of the
population
Sample
Conduct
survey/experiment to
collect data; organize and
summarize the data.
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