Chapter 8
Section 2
Distributions of the
Sample Proportion
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 1 of 13
Chapter 8 – Section 2
● Learning objectives
1

Describe the sampling distribution of a sample
proportion
2 Calculate probabilities of a sample proportion
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 2 of 13
Chapter 8 – Section 2
● Learning objectives
1

Describe the sampling distribution of a sample
proportion
2 Calculate probabilities of a sample proportion
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 3 of 13
Chapter 8 – Section 2
● In an election, polling companies wish to
estimate the percent of people who will vote for
each candidate
● This clearly is a situation for sampling as it is
impractical to contact every single voter
● The desired results are proportions, for example
that 59% of the voters (a proportion of 0.59) said
that they will vote for candidate A
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 4 of 13
Chapter 8 – Section 2
● We have the same questions for the sample
proportion as we had for the sample mean
 What is the mean of the sample proportion?
 What is the standard deviation of the sample
proportion?
 What is the distribution of the sample proportion?
 Can we apply the Central Limit Theorem to
approximate these with normal distributions?
● The answer is yes …
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 5 of 13
Chapter 8 – Section 2
● Sample proportions
● A random sample is take
 Of size n
 Each individual either has or does not have a certain
characteristic
 In total, there are x individuals that have this
characteristic
● Then the sample proportion pˆ (p hat) is given by
x
pˆ 
n
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 6 of 13
Chapter 8 – Section 2
● If a polling company polled 800 people to see if
they supported a certain issue and 475 did, then
we have a sample proportion problem with
 n = 800
 x = 475
 and a sample proportion of
475
pˆ 
 0.59
800
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 7 of 13
Chapter 8 – Section 2
● If the population proportion is p, then the
distribution of the sample proportion for a
sample of size n
 Is approximately normal if np(1-p) ≥ 10
 Has a mean of
 pˆ  p
 Has a standard deviation of
 pˆ 
p( 1  p )
n
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 8 of 13
Chapter 8 – Section 2
● Learning objectives
1

Describe the sampling distribution of a sample
proportion
2 Calculate probabilities of a sample proportion
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 9 of 13
Chapter 8 – Section 2
● Example
● Assume that 80% of the people taking aerobics
classes are female and a simple random sample
of n = 100 students is taken
 What is the probability that at most 75% of the sample
students are female?
 If the sample had exactly 90 female students, would
that be unusual?
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 10 of 13
Chapter 8 – Section 2
● The sample proportion pˆ of aerobics students
who are female
 Has an approximately normal distribution
 Has a mean of 0.80 and a standard deviation of 0.04
● What is the probability that pˆ is 0.75 or less?
 0.75 is 0.05 less than the mean of 0.80
 0.05 is 1.25 standard deviations less than the mean
(i.e. the z-score is –1.25)
 The normal probability P(z ≤ –1.25) = .1056
 Thus P( pˆ ≤ 0.75) = .1056
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 11 of 13
Chapter 8 – Section 2
● The sample proportion pˆ of aerobics students
who are female
 Has an approximately normal distribution
 Has a mean of 0.80 and a standard deviation of 0.04
● What is the probability that pˆ is 0.90 or more?
 0.90 is 0.10 more than the mean of 0.80
 0.10 is 2.5 standard deviations more than the mean
(i.e. the z-score is 2.5)
 The normal probability P(z ≥ 2.5) = .0062
 Thus P( pˆ ≥ 2.5) = .0062 … pretty unlikely
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 12 of 13
Summary: Chapter 8
● The sample proportion, like the sample mean, is
a random variable
 If the sample size n is sufficiently large and the
population proportion p isn’t close to either 0 or 1,
then this distribution is approximately normal
 The mean of the sampling distribution is equal to the
population proportion p
 The standard deviation of the sampling distribution is
equal to p( 1  p ) / n
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 13 of 13
Chapter 8 – Example 1
● According to a recent article, 98% of the people who travel on the New
Jersey Turnpike exceed the 60 miles per hour speed limit. (Source: Lange
JE, Johnson MB, Voas RB (2005) Testing the Racial Profiling Hypothesis for
Seemingly Disparate Traffic Stops. Justice Quarterly (22)2.) Law
enforcement officials are planning to covertly measure the speed of 1,000
cars on the Turnpike and will compute the proportion of drivers who exceed
the posted speed limit.
 a. Describe the sampling distribution of , the sample proportion of drivers who
exceed the speed limit.
 b. In a random sample of 1000 drivers, what is the probability that more than
97% of drivers will be speeding?
 c. Would it be unusual if in a random sample of 1000 drivers, 97% were
speeding?
 d. What is the minimum number of drivers that must be sampled to be sure that
is approximately normal?
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 14 of 13
Chapter 8 – Example 1
●
According to a recent article, 98% of the people who travel on the New Jersey
Turnpike exceed the 60 miles per hour speed limit. (Source: Lange JE, Johnson MB,
Voas RB (2005) Testing the Racial Profiling Hypothesis for Seemingly Disparate
Traffic Stops. Justice Quarterly (22)2.) Law enforcement officials are planning to
covertly measure the speed of 1,000 cars on the Turnpike and will compute the
proportion of drivers who exceed the posted speed limit.

a. Describe the sampling distribution of , the sample proportion of drivers who exceed the
speed limit. (Approximately normal with mean 0.98 and standard deviation 0.0044) pˆ

b. In a random sample of 1000 drivers, what is the probability that more than 97% of drivers
will be speeding? (0.9881)

c. Would it be unusual if in a random sample of 1000 drivers, 97% were speeding? (Yes,
since the probability that 97% or less would be speeding is 0.0119.)

d. What is the minimum number of drivers that must be sampled to be sure that is
approximately normal? (511)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 15 of 13
Chapter 8 – Example 2
● The quality control director of a small company that manufactures exercise
equipment knows that 10% of their products are defective. Market research
shows that only 30% of their customers will use the product in the first year
after the sale. So, the manufacturer expects that 3% of the units will be
returned for service under a one-year full warranty. This company expects to
sell 600 units next year. (This problem is based on data from an actual
company.)
 a. Describe the sampling distribution of , the proportion of the 600 units that will
be returned for service.
 b. What is the probability that more than 3% of the 600 units will be returned for
service?
 c. What is the probability that more than 5% of the 600 units will be returned for
service?
 d. Would it be unusual if in a random sample of 600 units, 5% were returned for
service?
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 16 of 13
Chapter 8 – Example 2
●
The quality control director of a small company that manufactures exercise
equipment knows that 10% of their products are defective. Market research shows
that only 30% of their customers will use the product in the first year after the sale.
So, the manufacturer expects that 3% of the units will be returned for service under a
one-year full warranty. This company expects to sell 600 units next year. (This
problem is based on data from an actual company.)

a. Describe the sampling distribution of , the proportion of the 600 units that will be returned
for service. (Approximately normal with mean 0.03 and standard deviation 0.006964) pˆ

b. What is the probability that more than 3% of the 600 units will be returned for service?
(0.5)

c. What is the probability that more than 5% of the 600 units will be returned for service?
(0.0020)

d. Would it be unusual if in a random sample of 600 units, 5% were returned for service?
(Yes)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 17 of 13