Download Sampling distributions • sample proportion ( ˆ p ) fraction (or

Chapter 18
1
Sampling distributions
• sample proportion ( pˆ )
fraction (or percentage) of measurements in a
sample that are “successes”:
€
pˆ = (# successes in sample) / n
• population proportion ( p )
€
fraction (or percentage) of the measurements in the
population that are “successes”; we also write
q = 1 − p to represent the fraction of “failures” in the
population
€
• sampling error/variability
variability in values of a statistic when it is
measured in different samples of the same size
randomly selected from the same population
• sampling distribution of a statistic
distribution of all possible values of the statistic
computed for every possible choice of (randomly
selected) sample of a fixed size n chosen from the
population
Chapter 18
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• sampling distribution model for pˆ
A SRS of size n with sample proportion pˆ is
selected from a population, chosen
so that it is
€
€
(1) small enough to be no more than 10% of the size
of the population (the 10% Condition), but
(2) large enough that it includes at least 10
successes and 10 failures (the Success/Failure
Condition),
then the normal model governs the sampling
distribution of values of pˆ . The random variable
represented by our measurements of pˆ has an
expected (mean) value
of E( pˆ ) = p, equal to the
€
population proportion, and a standard
deviation of
€
pq
€
SD( pˆ ) =
.
n
€
€
€
That is, the sampling distribution model for pˆ is

pq 
N  p,
.
n 

€
Moreover, this normal model is a better description
of the sampling distribution when the sample size n
€ is true because of…
is larger. This
Chapter 18
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the most fundamental theorem in all of statistical
theory…
• the Central Limit Theorem
Regardless of the population from which we are
sampling or the statistic we may be measuring, the
distribution of a sampling statistic comes closer to a
normal distribution the larger the sample size n
gets. Moreover, while the mean of the sampling
distribution does not depend on n, the standard
deviation does: it will decrease as n gets larger. So
the statistic we are sampling becomes a better
approximation of its true mean value for larger n.
Chapter 18
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• sampling distribution model for y
A SRS of size n with sample mean y is selected
from a population with population mean µ and
standard deviation σ. The sample is chosen so that
€
its measurements are independent of each other – a
€
condition that is difficult to check, but reasonable
to assume, unless the sample is drawn without
replacement, in which case it should be small
enough to be no more than 10% of the size of the
population (the 10% Condition).
Then the normal model governs the sampling
distribution of values of y . The random variable
represented by our measurements of y has an
expected (mean) value of E( y ) = µ , equal to the
population mean, and a standard deviation of
€
€
σ
€SD(
€y ) =
.
n
That is, the sampling distribution model for y is
€
€

σ 
N µ ,
.

n
€
Moreover, this normal model is a better description
of the sampling distribution the larger n gets.
€
Chapter 18
5
How large must the sample be to allow us to use
the normal model? This is hard to say in general;
but if the samples selected are themselves
unimodal and symmetric, the sampling distribution
will already be close to normal. For less symmetric
samples, larger sample sizes are warranted.
• standard error
In practice, we don’t generally know the value of
the standard deviation of the sampling distribution
of our statistic – after all, it depends on population
parameters: for instance,
SD( pˆ ) =
€
SD( y ) =
pq
n
σ
n
depends on p and q;
depends on σ.
€
But we can estimate the standard deviation. When
€ our estimates are called standard errors:
we do,
€
ˆpq
ˆ
;
n
s
SE( y ) =
.
n
SE( pˆ ) =
€
€
€
€