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Proportions
Suppose I have qualitative data. What can we do
with this type of data?
We create a “success” category that defines a
characteristic in which we are interested. We then
find the proportion of values that have this
characteristic, either for a sample or the entire
population. It is calculated as follows:
Proportion =
items with characteristic
number of items examined
Notation for Proportions
If we are talking about the entire population, the
proportion we find is the parameter and is given the
symbol p. Note that this is NOT Geometry, so the
value of the Greek letter Pi is NOT 3.14159…
If we are talking about a sample proportion, we use
the letter p.
Just as might use x to estimate m, we will use
the statistic p to estimate the parameter p.
Sampling Distribution of p
We need to know what the statistic’s sampling
distribution looks like (center, variability, normal?).
Suppose that 20% of all people in this area have
brown hair. (So p = 0.20)
A sample of 100 people is taken and 24 have brown
hair. So p1 = 24/100 = 0.24.
A second sample of 100 people is taken and 19
have brown hair. So p2 = 19/100 = 0.19.
Do this for 100,000 different samples.
Sampling Distribution of p
If I make a histogram of the 100,000 different
sample proportions (p1, p2, …, p100,000), what
patterns would I see?
1. Center: The mean of all sample proportions is
m p  p  0.20
Notice the “center” is the value of the parameter,
which makes sense. Some statistics will be higher
than the parameter, some will be lower. But on
average, they are centered at p.
Sampling Distribution of p
If I make a histogram of the 100,000 different
sample proportions (p1, p2, …, p100,000), what
patterns would I see?
2. Variability: (Keep 6 digits after the decimal point)
How different can the sample proportions be? The
reasons are beyond what we cover in the class, so
you just need to use the formula and “trust me!”
p 
p 1  p 
n
0.201  0.20

 0.0016  0.04
100
The Sampling Distribution of p
3.
The sampling distribution of the sample
proportion is approximately normal whenever
both of the following conditions are true:
i. p – 3
ii. p + 3
p 1  p 
n
p 1  p 
n
is more than 0
is less than 1
This ensures the distribution lies between 0 and 1.
The Sampling Distribution of p
We can then talk about probabilities if the third point
suggests that the sampling distribution is normal.
We create a z-score using the formula
z
p p
p 1  p 
n
Again, if the probability is less than .05, the value of
the sample proportion is unusual.
Example
A report claims that 68% of all Americans spent
$800 or more on gifts during the last holiday. A
random sample of 250 people in Cincinnati reveals
that 180 spent $800 or more for gifts.
A. Fully describe the sampling distribution of the
sample proportion for 250 people assuming the
report is true.
B. Would it be unusual to see the results observed
for the people of Cincinnati if the report is true?
Solution to Part A
1. Center
m p  p  0.68
2. Variability
0.681  0.68
p 
 .0008704  .029503
250
3. Normal?
i. 0.68 – 3(.029503) = .591 which is more than 0
ii. 0.68 + 3(.029503) = .769 which is less than 1
Since both conditions are satisfied, the
sampling distribution of p is normal.
Solution to Part B
Now p = 180/250 = 0.72 for the random sample.
Since .72 is more than .68 (the value of p), we want
to find the chance that any value of p is .72 or more.
.72  .68
First, convert .72 into a z-score. z 
 1.36
.029503
Now find P(Z > 1.36) = 0.5 – 0.4131 = .0869
Since .0869 is not less than .05, we do not consider
seeing 180 out of 250 people who spent more than
$800 to be unusual. We don’t doubt that p = 0.68.