Unformatted text preview: rtment of Statistics
G.
University of South Carolina; Slide 7 ˆ
Sampling Distribution for p
Sampling
ˆ
The sampling distribution of p based on
large n is approximately normal. Rule of Thumb: If np > 5 and n(1p) > 5
(preferably both > 10), then the distribution
of Y and hence Y/n can be approximated
with a normal distribution.
G. baker, Department of Statistics
G.
University of South Carolina; Slide 8 Sampling Distribution for ˆ
p To completely define the normal distribution of ˆ
p
We need the mean (expected value) and variance. G. baker, Department of Statistics
G.
University of South Carolina; Slide 9 ˆ
Sampling Distribution of p
Sampling 1α
p − zα / 2 p (1 − p )
n p p + zα / 2 p (1 − p )
n p(1 − p )
n ˆ
So, at most, p will be
away from p,
(1α)100% of the time. We call this (1α)100%
the level of confidence.
zα / 2 G. baker, Department of Statistics
G.
University of South Carolina; Slide 10 Repeated Sampling of Size n
95% 95% of the time
our estimate
will be within
p (1 − p )
1.96
n of the truth. G. baker, Department of Statistics
G.
University of South Carolina; Slide 11 Standard Error
We don’t know the value of p, so we will
We
know
use p
use ˆ ˆ
When we use p , we have an estimate of
the standard deviation for the sampling
ˆ
distribution of p .
distribution
We call this estimate the standard error. ˆ
ˆ
p (1 − p)
n G. baker, Department of Statistics
G.
University of South Carolina; Slide 12 Confidence Interval for p
ˆ
(1 − α )100%CI for p : p ± zα / 2 ˆ
ˆ
p(1 − p )
n Example: ˆ
ˆ
p (1 − p )
ˆ
95%CI for p : p ± 1.96
n
G. baker, Department of Statistics
G.
University of South Carolina; Slide 13 Confidence Interval Based on
Confidence
Normal Distribution
Normal
Pt Est ± ( Z Value) (Standard Error) ˆ
p ± zα / 2 ˆ
ˆ
p (1 − p )
n Standard error is our estimate of the standard
deviation for the distribution of the point
G. baker, Department of Statistics
G.
University of South Carolina; Slide 14
estimate. Confidence Interval Estimation
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 Fall '13
 Wang
 Statistics, Normal Distribution, Statistical hypothesis testing, G. Baker

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