Transcript Sampling Distributions PP

SAMPLING
DISTRIBUTIONS
. . . The
foundation of
statistical inference
procedures
Thanks to Ruth Reece, Jeanne Lorenson,
and Susan Blackwell
Before we talk about sampling
distributions, make sure you
understand the following:
PARAMETER – a measure for an
entire population, ex. 
STATISTIC – a measure for a
sample, ex. x
What is a Sampling Distribution?
A sampling distribution is the distribution of a
set of values of a sample statistic obtained
from all possible samples of a given size
from a given population.
Like any distribution, a sampling distribution
can be described by its mean, standard
deviation, and shape.
A statistic is unbiased if . . .
. . .the mean of the sampling distribution of
the sample statistic equals the value of the
population parameter being estimated.
Examples:
 pˆ  p
sample medians  median of the population
X  X
Variability of Sampling
Distributions
The larger the sample size of the
samples, the less variable the sampling
distribution will be.
Format for Problems with
Sampling Distributions
1. Get mean of sampling distribution based
on having SRS.
2. Get standard deviation of sampling
distribution based on population > 10n.
3. Establish if normal (will depend on what
you are sampling).
4. Calculate probability.
Sample Means
1. If the sample results from an SRS, then  X   X .
X
2. If the population is  10n, then  X 
.
n
3. a. If the populatin distribution is normal,
then the X distribution is normal.
b. If the population distribution is approximately normal,
then the X is approximately normal.
c. If the population distribution is nonnormal or you don't
know the shape and the sample size is large,
then the X distribution is APPROXIMATELY normal.
(This is based on the Central Limit Theorem.)
4. Calculate the probability based on the  X and  X .
Central Limit Theorem
The CLT is about SHAPE.
It says that the sampling
distribution of sample means
becomes more closely normal in
shape as the sample size
increases.
What is Large?
If n is 30 or larger, sample is large
enough for distribution of means
to be approximately normal.
(Some books say 40, but 30 is ok)
If n is larger than 15 with no
outliers or apparent skewness,
sample is large enough for
distribution of means to be
approximately normal.
Suppose that the heights of Va. Tech students are
normally distributed with a mean of 65 inches and a
standard deviation of 2.5 inches.
1. What is the probability that a randomly selected student
is taller than 5.5 ft tall?
.344
2. What is the probability that the mean height of 30
students is greater than 5.5 ft?
.014
3. Could you do #1 if the heights were not normally
distributed?
No
4. Could you do #2 if the heights were not normally
distributed?
Yes
Sample Proportions
1. If the sample results from an SRS, then  pˆ  p.
2. If the population is  10n, then  pˆ 
p 1  p 
.
n
3. If np  10 and n 1  p   10, then the pˆ distribution
is APPROXIMATELY normal.
(Some books say 5 instead of 10.)
4. Calculate the probability based on the  pˆ and  pˆ .
Which of the following are true?
I. Sample parameters are used to make
inferences about population statistics.
II. Statistics from smaller samples have
more variability.
III. Parameters are fixed, while statistics vary
depending on which sample is chosen.
Answer: II and III (#I – no such thing as a
sample parameter!)