Sampling Distributions
Chapter 7
The German Tank Problem
In WWII, the Allies captured several German
Tanks. Each one had a serial number on it.
The German Tank Problem
Allied commanders wanted to know how many
tanks the Germans actually had! They sent
the serial numbers to a group of
mathematicians in Washington D.C.
The German Tank Problem
The mathematicians produced remarkably close
estimates for the actual number of German
Tanks.
http://en.wikipedia.org/wiki/German_tank_pro
blem
The Textbook Problem
How many Algebra 2 books are in this school?
The Textbook Problem
• Random Sample:
– 10
– 38
– 59
– 61
– 74
– 90
– 94
Sampling Distribution
• Samples are used to find out about the whole
population.
• A statistic is a number that describes the
__________.
• A parameter is a number that describes the
______________.
• In statistics, the value of a parameter is usually
unknown because
Notation
• Sample mean:
• Population mean:
• Sample proportion:
• Population proportion:
Example 1
• A pediatrician wants to know the 75th
percentile for the distribution of heights for 10
year old boys, so she takes a sample of 50
patients and calculates Q = 56 inches.
– Population
– Sample
– Parameter
– Statistic
Example 2
• A poll asked 1,102 12-17 year olds in the
United States if they have a cell phone. Of the
respondents, 71% said yes.
– Population
– Sample
– Parameter
– Statistic
Colored Chips
• Take out a handful of 20 chips. If you get more
just randomly toss some back.
• Count the number of RED CHIPS and record it
on your index card.
• What proportion of your chips were red?
• Example: 8 red chips/20 total = 0.4
• Record this number on your index card, too.
• Shuffle and pass to the next person.
Colored Chips
Number of RED =
Proportion of RED =
.
20
= ___
Sampling Variability
• Each time we sample, we will get slightly
different information. Why?
• Because of this, it is best to take a large
number of samples from the same population.
– Calculate the same statistic for every sample.
– Make a graph of these values.
– Examine the distribution (SOCS)
– Activity: Sampling
Sampling Distribution
• The distribution of values taken by the statistic
in all possible samples of the same size from
the same population.
• If you were trying to find the mean age of
people in Tennessee, the sampling distribution
would be the list of all the means from every
sample that could be taken.
Sampling Distribution
• It’s too difficult to take ALL POSSIBLE SAMPLES
• Sometimes we use simulations to imitate this
process.
• If you don’t use ALL POSSIBLE SAMPLES, it’s
not a sampling distribution! It would be an
approximation.
Sampling Distribution
• In the activity you are working on (colored
chips), we cannot make an actual sampling
distribution.
• We are limited to taking many samples. We
will take about 20, but ideally we would take
many more.
• I will use your samples to create a dotplot.
Proportion of RED chips
10
9
8
7
6
5
4
3
2
1
0
0
0.1
0.2
0.3
0.4
p
0.5
0.6
0.7
0.8
Sampling Distribution
• Once we see the dotplot, we will analyze:
– Shape
– Center
– Spread
– Outliers
SOCS
•
•
•
•
Shape: symmetry? Skewed left/right?
Center: Balance Point, Mean?
Spread: Variability, Standard Deviation?
Outliers: Any outliers or unusual features?
Assignment
• On a separate sheet of paper, answer the 3
Check Your Understanding Questions on page
420-421.
Unbiased Estimator
• How well does the sample proportion
estimate the real proportion of red chips?
• Remember, the real proportion is 0.5
• If the mean of the sampling distribution = the
true value of the parameter being estimated,
the statistic is called an unbiased estimator.
Sample Proportions
7.2
The sampling distribution of p
• Shape:
• Center:
• Spread:
The sampling distribution of p
• Shape: in some cases, it looks like a normal
curve. This depends on n and p.
• Center: The mean = p. This is because p is an
unbiased estimator.
• Spread: The standard deviation gets smaller as
n gets larger. The value of it depends on n and
p.
Sample Proportion
p=
Mean and Standard Deviation
CYU page 437
Using the Normal Approximation