Lecture 12, Section 8.1 & 8.2
Proportions
Many statistical studies produce counts rather than measurements.
Example:
Did you vote in the last election?
The response would be either a
“Yes” or a “No”.
The variable is categorical, the response is the value the
variable takes on for each unit/person.
If I did a survey of this class, I could
accumulate the count of “Yes” responses and describe this count as a
proportion of the total.
Example:
What academic year are you in at Purdue.
The response would
be either “Freshman”, “Sophomore”, “Junior”, or “Senior”.
Again, I could
accumulate the count of each and describe each as a proportion of the total.
Population and Sample proportions:
In statistical sampling we often want to estimate the proportion,
p,
of
“successes” in a population.
“Success” is when the categorical variable
takes on one particular value.
p
=
count of successes in population
size of population
=
X / N
We take a sample of our population; our estimator is the sample proportion
of successes:
¶
p
=
count of successes in sample
size of sample
=
X
/
n
Example:
1.
You flip a coin 20 times and record whether a head or a tail is tossed.
In
this sample, a head is recorded 11 times.
What is the sample proportion
of heads?
Lecture 12, Section 8.1 & 8.2
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Inference for a Single Proportion:
So far we have only looked at making statistical inference on population
means, a measurement of some quantitative variable of interest.
Now we
will look at making statistical inference on a categorical variable using the
proportion of some outcome/success in a population.
Examples:
•
How common is it for students at Purdue to fail a class?
Out of a
sample of 200 students, 50 of them have failed at least one class, or
25% .
Based on these data, what can we say about all students at
Purdue?
•
What proportion of golfers in the USA have made at least one holein
one in their lifetime.
From an SRS of 50 golfers 25 of them had made
a holeinone.
What can we say about all golfers in the USA?
In both examples above we are interested in estimating the unknown
proportion
p
from a population.
The estimate of that population parameter
p
is the sample proportion
µ
p
, a statistic.
Sampling Distribution of a Sample Proportion:
Choose an SRS of size
n
from a large population with population proportion
p
having some characteristic of interest.
We normally call whatever
characteristic we are studying a “success.”
Let X be the count of successes
in the sample and
µ
p
be the sample proportion of success,
µ
p
= X/
n
Also:
•
The sampling distribution of
µ
p
is approximately normal for a SRS
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 Spring '08
 Staff
 Normal Distribution, Statistical hypothesis testing, µ, p1 − p2

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