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Sampling Distribution Models
Population – all items of interest.
Sample – a few items from the population.
Population Parameter: p
Sample Statistic:
Inference
Random selection (^) p ˆ
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Sampling Distribution of
Shape: Approximately Normal
Center: The mean is p.
Spread: The standard deviation
is
p ˆ
( )
n
p 1 − p
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Sampling Distribution of
Conditions:
- 10% Condition: The size of the sample should be less than 10% of the size of the population.
- Success/Failure Condition: n p and
n(1 – p ) should both be greater than
p ˆ
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n p pq n p pq n pp pq n p pq n p pq n p − 3 pq − 2 − + 1 + 2 + 3
68 – 95 – 99.7 Rule
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Probability
If the population proportion, p ,
is known, we can find the
probability or chance that
takes on certain values using a
normal model.
p ˆ
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Inference
In practice the population parameter,
p , is not known and we would like to
use a sample to tell us something
about p.
Use the sample proportion, , to
make inferences about the population
proportion p.
p ˆ
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68-95-99.7 Rule
95% of the time the sample
proportion, , will be within
two standard deviations of p.
p ˆ
n
p ( 1 p ) 2
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Standard Deviation
Because p , the population proportion
is not known, the standard deviation
is also unknown.
n
p p p
SD( ˆ)
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Standard Error
Substitute as our estimate
(best guess) of p.
The standard error of is:
p ˆ
p ˆ
n
p p SE p
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About 95% of the time the sample
proportion, , will be within
two standard errors of p.
p ˆ
n
p p SE p
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About 95% of the time the sample
proportion, p , will be within
two standard errors of.
n
p p SE p
ˆ( 1 ˆ) 2 (ˆ) 2
−
p ˆ
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Confidence Interval for p
We are 95% confident that p
will fall between
n
p p p n
p p p
andˆ 2
