NOTES: Sampling Distributions
n
= sample size
x
= sample mean = average of a quantitative variable describing a SAMPLE
µ
= population mean = average of a quantitative variable describing the POPULATION
p
ˆ
= proportion of the SAMPLE with trait
n
traithave which SAMPLE in the sindividual ofnumber
=
p
= proportion of the POPULATION with trait
sizepopulation
traithave which POPULATION in the sindividual ofnumber
=
Statistic: a number which describes a sample.
Examples:
x
,
x
S
,
p
ˆ
,
n
Parameter: a number which describes a population.
Examples:
µ
,
σ
,
p
Sampling Distribution: The sampling distribution of a statistic is the distribution of values taken by the
statistic in all possible samples of the same size from the same population.
Unbiased Estimator: A statistic used to estimate a parameter is unbiased if the mean of its sampling
distribution is equal to the true value of the parameter being estimated.
Example:
(
)
pp =
ˆ
µ
Example:
(
)
µµ
=x
NOTES: z-scores for distributions
In general, the z-score for a value in a sampling distribution is the value minus the mean of the distribution
divided by the standard deviation of the distribution. In notation:
deviation
standard
meanvalue
−
. This concept is
the origin for the different sampling distribution z-score formulas seen in the table below.
Distribution z-score z-score with
replaced formulas
x
σ
µ
−
x
σ
µ
−
x
x
(
)
( )
xSD
xx
µ
−
n
x
σ
µ
−
p
ˆ
(
)
( )
pSD
pp
ˆ
ˆˆ
µ
−
( )
n
pp
pp
−
−
1
ˆ
