Download Sampling distributions

Sampling distributions
Example
• Take random sample of students.
• Ask “how many courses did you study for
this past weekend?”
• Calculate a statistic, say, the sample mean.
Sample 1:
1
0
2
Mean = 1.0
Sample 2:
1
1
4
Mean = 2.0
Situation
• Different samples produce different results.
• Value of a statistic, like mean or proportion,
depends on the particular sample obtained.
• But some values may be more likely than
others.
• A “sampling distribution” is a probability
distribution of a statistic. It indicates the
likelihood of getting certain values.
Sampling distribution of mean
IF:
• data are normally distributed with mean 
and standard deviation , and
• random samples of size n are taken, THEN:
The sampling distribution of the sample means is
also normally distributed.
The mean of all of the sample means is .
The standard deviation of the sample means
(“standard error of the mean”) is /sqrt(n).
Example
• Adult nose length is normally distributed
with mean 45 mm and standard deviation 6
mm.
• Take random samples of n = 4 adults.
• Then, sample means are normally
distributed with mean 45 mm and standard
error 3 mm [from 6/sqrt(4) = 6/2].
Using empirical rule...
• 68% of samples of n=4 adults will have an
average nose length between 42 and 48 mm.
• 95% of samples of n=4 adults will have an
average nose length between 39 and 51 mm.
• 99% of samples of n=4 adults will have an
average nose length between 36 and 54 mm.
What happens if we take larger
samples?
• Adult nose length is normally distributed
with mean 45 mm and standard deviation 6
mm.
• Take random samples of n = 36 adults.
• Then, sample means are normally
distributed with mean 45 mm and standard
error 1 mm [from 6/sqrt(36) = 6/6].
Again, using empirical rule...
• 68% of samples of n=36 adults will have an
average nose length between 44 and 46 mm.
• 95% of samples of n=36 adults will have an
average nose length between 43 and 47 mm.
• 99% of samples of n=36 adults will have an
average nose length between 42 and 48 mm.
• So … the larger the sample, the less the
sample averages vary.
What happens if data are not
normally distributed?
The Central Limit Theorem tells us ...
Central Limit Theorem
• Even if data are not normally distributed,
as long as you take “large enough” samples,
the sample averages will at least be
approximately normally distributed.
• Mean of sample averages is still .
• Standard error of sample averages is still
/sqrt(n).
• In general, “large enough” means more than
30 measurements.
Big Deal?
• Knowing the distribution of sample means
allows us to make decisions about the value
of a population mean.
• Let’s look at an application …