Section 9.1 Estimating a Population Proportion
Objective: Obtain a point estimate for the population proportion; construct and
interpret a confidence interval for the population proportion; determine
sample size necessary for estimating a population proportion.
A point estimate of a population characteristic is a single number that is based on sample
data and estimates the value of a parameter.
 x is used as a point estimate for .
 s is used as a point estimate for .

p̂ is the point estimate for p. pˆ 
x
n
A confidence interval for an unknown parameter is an interval of possible values for the
parameter. It is constructed so that, with a chosen degree of confidence, the actual value
of the parameter will be between the lower and upper bounds of the interval.
A level of confidence, c, is a measure of the degree of assurance we have in our point
estimate. The confidence level associated with a confidence interval is the success rate of
the method used to construct the interval.
 Theoretically, the value of c may be any number between zero and one.
 Typical values for c include 0.90, 0.95, and 0.99.
 z  is called the critical value for a confidence level c. (Remember, z is the z-score
2
such that the area under the standard normal curve to the right of z is , so use
the positive z-score.)
o The determine the critical value for a given level of confidence,
1 c

z   InvNorm 
, 0, 1  . For example, if given a 95% level of confidence,
 2

2
 1  0.95

the critical value will be z   InvNorm 
, 0, 1   1.96
2


2
 Confidence interval estimates for the population proportion are of the form
point estimate  margin of error
Confidence Interval for p
pˆ  E  p  pˆ  E
9.1 - 2
The margin of error, E, determines the width of the interval and depends on :
 Level of confidence – as the level of confidence increases, the margin of error also
increases.
 Sample size – as the sample size increases, the margin of error decreases (Law of
Large Numbers)
 Standard deviation – an increase in the standard deviation will widen the confidence
interval.
Constructing a Confidence Interval about p
In order to construct a confidence interval for a population proportion, the following
condition must be met:
ˆ
 npˆ (1  pˆ )  10 where p
x
and the sample size is no more than 5% of the
n
population size.
We will use the TESTS menu on the graphing calculator to construct our confidence
intervals.
Using TI 83/84 to Construct Confidence Intervals about p




Press STAT, highlight TESTS, and select A:1-PropZInt
Enter the values of x and n
Enter C-Level:
Press Enter on Calculate
Interpretation of a Confidence Interval for p –
“We are _____% confident that the population proportion is between (lower bound) and
(upper bound). ”
OR
“We are _____% confident that the interval actually does contain the true value of p.”
Work #1 - 4
9.1 - 3
Determine the Sample Size Necessary for Estimating a population Proportion
Compute Minimum Sample Size
We know E  z 
2
pˆ (1  pˆ )
n
 z

Solving for n, we get n  pˆ (1  pˆ )  2
 E






2
(round z to three decimal places)
Any fractional value of n is always rounded to the next higher whole number.
Compute Minimum Sample Size
 If we have no preliminary estimate for p
 z

n  .25 2
 E






2
(round z to three decimal places)
Any fractional value of n is always rounded to the next higher whole number.
Work #5 - 6