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Sampling Variability and Sampling
Distributions
How can a sample be used to accurately present the
characteristics of a population?
Statistic is any quantity computed from values in a sample
Sampling distribution the distribution of one statistic over
many samples often look at histograms to determine
patterns
Sampling variability is the recognition that the value of a
statistic depends on the sample selected , the size of the
sample and how many samples are evaluated
Since: x  
x  
x 

n
since the average of the samples
approximates the pop ave.
Consider a population consisting of the following five values {8, 14, 16,10, 11},
which represent the number of video rentals during the academic year for
each of the five housemates.
a) compute the mean of this population
b) select a random sample of size 2 by writing the numbers on slips of paper
and selecting two. Compute the mean of the sample.
c) repeatedly select samples of size two, and compute the associated values
until you have looked at 25 samples
d) construct a histogram using the 25 samples. Are most of the values near
the population mean? Do they differ a lot from sample to sample or do they
tend to be similar?
Homework page 409-411 evens
For #12 the info is on pages 406 and 409
If
is based on a large # of items (n) in the sample the x is
closer to µ than if n is small ( although x does approximate
µ regardless of the size of n)
If
n is large σ is closer to Sx (meaning the data is less
variable)
x
Rules of the Sampling Distribution
1)  x  
2)   
x
(correct as long as no more than 5% of the population is included)
n
3) when the population is normal, the sampling distribution
of x is also normal for any sample size of n
4) When n is sufficiently large (ie approx 30+) the sampling
distribution of x is approximated by a normal curve even
when the population is not.
(known as the CENTRAL LIMIT THEOREM)
If n is large or the population distribution is
normal
z
x
x  x
x

x

n
is the standard deviation of the sample
averages
Example 8.7 page 418
A soft-drink company claims that on average, cans
contain 12 oz. of soda. Let x denote the actual volume of
soda in a randomly selected can. Suppose that x is
normally distributed with σ= .16 oz. Sixteen cans are to
be selected and the soda volume determined for each one.
Let x denote the resulting sample average soda volume.
Because the x distribution is normal, the sampling
distribution is normal.
If the bottler’s claim is correct, the sample distribution of
has a mean value     12 and a standard deviation of
_
 
_
x

n
x

.16
16
 .04
a) determine the probability that a can of soda contains between 11.96 oz and
12.08 oz
Z-chart
b) if you choose a can of soda at random, what is the probability that
the manufacturer is correct if your soda has at least 12.1 oz in it.
HW Pg 420 to 422 8.14 to 8.22 even
The Sample proportion p = number of successes
RULES:
1) µp = π
2) σp =  (1   )
# in the sample(n)
n
3) when n is large and π is not too near to 0 or 1 the sampling
distribution of p is approximately normal
--the further π is from .5 the larger n must be for the
sampling x distribution to be approx normal.
Rule of thumb:
conservative
liberal
If nπ ≥ 10 and n(1-π) ≥10
nπ≥5 and n(1-π)≥5
Note: the book tends to use the conservative method and the AP test tends to use
the liberal method according to the AP listserve it doesn’t make a difference just
show your set up!!!!!
AP Statistics Example 8.11 page 426
The proportion of all blood recipients stricken with viral hepatitis was given as .07 in an
article in the American Journal of medicine. Suppose a new treatment is believed to
reduce the incidence rate of viral hepatitis. This treatment is given to 200 blood recipients.
Only 6 of the 200 contracted hepatitis. It appears that the new treatment is better since p =
6/200 = .03.
Determine if the sampling distribution of p is normally distributed
Determine the probability that the sample distribution p could be less than .03 if the
treatment were ineffective (i.e. it did not change  --the original value of those
contracting viral hepatitis under the old treatment)
Hw pg 427 8.28 to 8.32 even
Formulas:
Distribution of
x
x  
x 

n
If n is large the x dist is
Normal even if the pop isn’t
Distribution of p
1) µp = π
2) σp =  (1   )
n
The further π is from .5
the larger n must be for the
distribution to be considered
normal so:
show the conservative or
liberal test
Homework pg 428 evens
Test set up:
9 multiple choice—1 point each
3 computational—total of 45 points
Test total 54 points