Unformatted text preview: with a standard
deviation of 300 hours.)
deviation
If we randomly sample 1 bulb, will we realize
that the average life has decrease? What if we
sample 3 bulbs? 9 bulbs?
sample
G. Baker, Department of Statistics
G.
University of South Carolina; Slide 9
University Why Averages Instead of Single
Why
Readings?
Readings?
µ = 1500 800 1300 σ = 300 µ = 2000 1800 2300 2800 Single Readings
Y < 1400 would signal shift G. Baker, Department of Statistics
G.
University of South Carolina; Slide 10
University
10 Why Averages Instead of Single
Why
Readings?
Readings?
µ = 1500 800 1300 σ = 173 µ = 2000 1800 2300 2800 Averages of n = 3
Y < 1654 would signal shift G. Baker, Department of Statistics
G.
University of South Carolina; Slide 11
University
11 Why Averages Instead of Single
Why
Readings?
Readings?
µ = 1500
µ = 1500
µ = 1500 800 1300 µ = 2000
µ = 2000
µ = 2000 1800 2300 σ = 100 2800 Averages of n = 9
Y < 1800 would signal shift G. Baker, Department of Statistics
G.
University of South Carolina; Slide 12
University
12 What if the original distribution
What
is not normal? Consider the roll
of a fair die:
of
Rolling A Fair Die Probability 0.20
0.15
0.10
0.05
0.00
1 2 3 4 5 6 # of Dots
G. Baker, Department of Statistics
G.
University of South Carolina; Slide 13
University
13 Suppose the single measurements
Suppose
are not normally Distributed.
Let Y = time to fail of a light bulb in
constant failure rate mode
constant
Y is exponentially distributed
is
with λ = 0.0005 = 1/2000
with
0.0005 0 1000 2000 3000 4000 5000 6000 G. Baker, Department of Statistics
G.
8000
University of South Carolina; Slide 14
University
14 7000 Single measurements Averages of 2 measurements Averages of 4 measurements Averages of 25 measurements Source: Lawrence L.
Lapin, Statistics in
Modern Business
Decisions, 6th ed.,
1993, Dryden Press,
Ft. Worth, Texas. G. Baker, Department of Statistics
G.
University of South Carolina; Slide 15
University
15 n=1 As n increases, what
happens to the variance? n=2 n=4 A.Variance increases.
B.Variance decreases.
C.Variance remains the
same. n=25 G. Baker, Department of Statistics
G.
University of South Carolina; Slide 16
University
16 n=1
n=2
n=4 n = 25 G. Baker, Department of Statistics
G.
University of South Carolina; Slide 17
University
17 Central Limit Theorem
If n iis sufficiently large, the sample
If s
means of random samples from a
population with mean µ and standard
and
deviation σ are approximately normally
are
distributed with mean µ and standard
and
deviation σ / n .
deviation G. Baker, Department of Statistics
G.
University of South Carolina; Slide 18
University
18 Random Behavior of Means Summary
If Y is distributed n(µ, σ), then yn
n(
),
is distributed n(µ, σ / n ).
n(
If Y is distributed nonn(µ, σ), then y x≥30
If
),
is distributed approximately
n(µ, σ / n ).
n(
).
G. Baker, Department of Statistics
G.
University of South Carolina; Slide 19
University
19 If We Can Consider y to be Normal …
If
Recall: If Y is distributed normally with
mean µ and standard deviation σ, then
and
Y −µ
Z= σ So if y is distributed normally with
mean µ and standard deviation σ / n ,
then
then Y −µ
Z=
σ/ n G. Baker, Department of Statistics
G.
U...
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 Fall '13
 Wang
 Statistics, Normal Distribution, Standard Deviation, G. Baker

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