STATISTICS
CHAPTER 8
Confidence Intervals
of the Population Proportion
Presentation Goals
The goal of this presentation is that you will have familiarity
and an understanding with the following:
• Confidence Intervals of the Population Proportion
• Standard Normal Distribution
• NORM.S.DIST() & NORM.S.INV()
Sample vs. Population
Recap Notation for Proportion
Sample Statistics:
n = sample size
x = number of successes
𝑝=
𝑥
𝑛
= sample proportion
(we say ‘p-hat’)
qˆ  1.00  pˆ
Population Parameter:
p = population proportion
q = 1.00 - p
Single-point estimates:
vs. Confidence Interval
The best single-point estimate of the population proportion, p, is
the sample proportion, 𝑝.
However, in estimating the population proportion we hope to know
how accurate our estimate is, and for that we turn to the idea of
constructing of a confidence interval for the population proportion.
Confidence Intervals:
of the population proportion, p
The confidence interval for the population proportion is centered around the
sample proportion, 𝑝, and extends the same distance in either direction, the
margin of error, E.
There are several common notations:
± notation: 𝑝 ± 𝐸
interval notation: (𝑝 − 𝐸 , 𝑝 + 𝐸)
using inequalities: 𝑝 − 𝐸 < p < 𝑝 + 𝐸
C.I. of the Population Proportion:
Assumptions
We will use probability distributions to find our margin of error, and
that process will determine the confidence level.
We make a
•
•
•
few assumptions:
We have a simple random sample
The conditions of a binomial distribution are met
𝑝n ≥ 5 & 𝑞n ≥ 5 (where 𝑞 = 1 - 𝑝)
These conditions allow us to apply the Central Limit Theorem.
We generally assume the first two and check that the third condition is satisfied.
Margin of Error:
C.I. of Population Proportion
Confidence Interval of the Population Proportion:
± notation: 𝑝 ± 𝐸
interval notation: (𝑝 − 𝐸 , 𝑝 + 𝐸)
using inequalities: 𝑝 − 𝐸 < p < 𝑝 + 𝐸
The Error is given by:
Common 𝑧𝛼 2 values:
E  z /2 
pˆ  qˆ
n
90% C.I., 𝑧𝛼 2 = 1.645
95% C.I., 𝑧𝛼 2 = 1.96
99% C.I., 𝑧𝛼 2 = 2.576
(we say, “z alpha over 2”)
C.I. of Population Proportion:
Example
Example:
A survey of 345 randomly selected students found that 301 students think that
parking is too expensive on campus. Construct and interpret the 90% confidence
interval for the proportion of all students (the population) who think parking is
too expensive on campus.
C.I. of Population Proportion:
Example
Solution:
n = 345
x = 301
Find 90% C.I. of population proportion:
Find the sample proportion:
pˆ 
x
301

 0.872464
n
345
&
qˆ  1  pˆ 
44
 0.127536
345
(Use exact fractions in later computations or round to at least six decimal
places until after we find the error. At the end of the problem, round
according to the individual problem’s instructions.)
C.I. of Population Proportion:
Example
Solution:
n = 345, x = 301, Find 90% C.I. of p (𝑧𝛼 2 = 1.645):
pˆ 
x
301

 0.872464
n
345
&
qˆ  1  pˆ 
Calculate the margin of error:
E  z /2 
44
 0.127536
345
 301  44 



pˆ  qˆ
345  345 

 1.645 
 0.0295
n
345
Construct the confidence interval (choose one):
p ± E : 0.8724 ± 0.0295
(p − E , p + E) : (0.843, 0.902)
p − E < p < p + E : 0.843 < p < 0.902
We are 90% confident that the population proportion of students who think parking on
campus is too expensive is between 84.3% and 90.2%.
C.I. of Population Proportion:
Example, (Find 𝑧𝛼 2 )
Example:
A survey of 220 randomly selected registered voters found that 132 of them voted
in the midterm election. Construct and interpret the 80% confidence interval for
the proportion of all registered voters (the population) who voted in the midterm
election.
C.I. of Population Proportion:
Example
Solution:
n = 220
x = 132
𝑝 = x/n = 132/220 = 0.60
80% Confidence Interval of p:
Here we must first find 𝑧𝛼 2 , because the 80% confidence level is not one
of the common values (90%, 95%, 99%) where we know 𝑧𝛼 2 .
C.I. of Population Proportion:
Example
Solution:
n = 220
x = 132
𝑝 = 0.60
80% C.I. of p
𝑧𝛼 2
= NORM.S.INV(area to the left of 𝑧𝛼 2 )
= NORM.S.INV(0.90)
= 1.28
(1.00 – 0.80)/2 = 0.10. Add this to 0.80 to get area to the left of 𝑧𝛼 2 , then use
NORM.S.INV().
𝛼
2=
Find 𝑧𝛼 2 : NORM.S for standard normal distribution
C.I. of Population Proportion:
Example
Solution:
n = 220, x = 132, 𝑝 = 0.60, 80% C.I. of p, 𝑧𝛼 2 = 1.28
Find Error:
E  z /2 
pˆ  qˆ
 1.28 
n
 0.60  0.40   0.042
220
Construct the confidence interval (choose one):
𝑝± E
: 0.60 ± 0.042
(𝑝 - E , 𝑝 + E ) : (0.558 , 0.642)
𝑝 - E < p < 𝑝 - E : 0.558 < p < 0.642
We are 80% confident that the proportion of registered voters who voted in the midterm
elections is between 55.8% and 64.2%.
C.I. of Population Proportion:
Recap
To construct a confidence interval of the population proportion:
𝑝- E< p <𝑝+E
where
E  z /2 
pˆ  qˆ
n , 𝑝 = x/n and 𝑞 = 1 - 𝑝
Commonly used 𝑧𝛼 2 ’s:
90% C.I., 𝑧𝛼 2 = 1.645
Uncommonly used 𝑧𝛼 2 ’s:
𝑧𝛼 2 = NORM.S.INV(area to the left of 𝑧𝛼 2 )
95% C.I., 𝑧𝛼 2 = 1.96
where area to the left of 𝑧𝛼
99% C.I., 𝑧𝛼 2 = 2.576
confidence level + alpha/2
2
is found by taking