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Sampling Distributions for Mean and Median mean
median 8 6 4 2 0 2 4 6 8 For normal populations, the sample mean is the most efficient
G. Baker, Department of Statistics
G.
estimator of µ.
University of South Carolina; Slide 39
University
39 Interval Estimate of the Mean
Yn − µ
Z=
follows a standard normal distribution
σ/ n Y −µ
P (−1.96 <
< +1.96) = 0.95
σ/ n P( µ = Y ± 1.96 σ
n (with a little algebra) ) = 0.95 So we say that we are 95%
confident that µ is in the interval Y ± 1 . 96 σ
n What assumptions have we
made?
G. Baker, Department of Statistics
G.
University of South Carolina; Slide 40
University
40 Interval Estimate of the Mean
Standard Normal 0.95 .025 4 3 2 1.96 1 0 .025
1 2 1.96 3 4 Z G. Baker, Department of Statistics
G.
University of South Carolina; Slide 41
University
41 Interval Estimate of the Mean
Let’s go from 95% confidence to the
Let go
general case.
general
The symbol zα is the zvalue that has an
value
area of α to the right of it.
P(− zα / 2 Y −µ
<
< + zα / 2 ) = (1 − α )
σ/ n P ( µ = Y ± zα / 2 σ
n ) = (1 − α )
G. Baker, Department of Statistics
G.
University of South Carolina; Slide 42
University
42 Interval Estimate of the Mean
Standard Normal 1α α/2 4 3 Zα/2 2 1 0 α/2 1 +Zα/22 3 4 (1 – α) 100% Confidence Interval
G. Baker, Department of Statistics
G.
University of South Carolina; Slide 43
University
43 What Does (1 – α) 100% Confidence Mean?
What
Sampling Distribution
of the y n( µ , σ / n)
y y Z
y 8
7 S mle
ap x y 6
5 y 4
3
2 y y x y
y 1 xy
x (1α)100%
Confidence
Intervals 0 µ
G. Baker, Department of Statistics
G.
University of South Carolina; Slide 44
University
44 If Z0.05 = 1.645, we are _____%
1.645,
confident that the mean is between y ± 1.645 σ
n A.99%
B.95%
C.90%
D.85% G. Baker, Department of Statistics
G.
University of South Carolina; Slide 45
University
45 Which zvalue would you use to
value
calculate a 99% confidence
interval on a mean?
interval
Z0.10 = 1.282
B. Z0.01 = 2.326
C. Z0.005 = 2.576
D. Z0.0005 = 3.291 A. G. Baker, Department of Statistics
G.
University of South Carolina; Slide 46
University
46 Plastic Injection Molding Process
A plastic injection molding process for a
part that has a critical width dimension
historically follows a normal distribution
with a standard deviation of 8.
with
Periodically, clogs from one of the feeder
lines causes the mean width to change. As
a result, the operator periodically takes
random samples of size 4.
random
G. Baker, Department of Statistics
G.
University of South Carolina; Slide 47
University
47 Plastic Injection Molding
A recent sample of four yielded a sample
mean of 101.4.
mean
Construct a 95% confidence interval for
the true mean width.
the
Construct a 99% confidence for the true
mean width.
mean G. Baker, Department of Statistics
G.
University of South Carolina; Slide 48
University
48 When going from a 95% confidence
When
interval to a 99% confidence interval,
the width of the interval will
the
Increase.
B. Decrease.
C. Remain t...
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 Fall '13
 Wang
 Statistics, Normal Distribution, Standard Deviation, G. Baker

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