Section 10.1 - Comparing Two Proportions (pp. 609-629)
Background - Often times we want to compare proportions of individuals with a certain characteristic
from two different populations. This might be the result of sampling two distinct populations or
comparing the effectiveness of two distinct treatments in an experiment.
The Sampling Distribution of a Difference between Two Proportions
From Chapter 7, we saw that the sampling distribution of ^p has the following properties:
Shape:
Center:
Spread:
We can use the formulas for combining two independent random variables to describe the sampling
distribution of ^p1− ^p2 :
Mean:
Standard Deviation:
The Sampling Distribution of ^p1− ^p2
Choose an SRS of size n1 from Population 1 with proportion of successes p1 and an independent SRS
of size n2 from Population 2 with proportion of successes p2.
Shape:
Center:
Spread:
, Example – Nathan and Kyle both work for the Department of Motor Vehicles (DMV), but in different
states. In Nathan’s state, 80% of the registered cars are made by American manufacturers. In Kyle’s
state, only 60% of the registered cars are made by American manufacturers. Nathan selects a random
sample of 100 cars from his state and Kyle selects a random sample of 70 cars from his state. Let
^p N − ^p K be the difference in the sample proportion of cars made by American manufacturers.
a. What is the shape of the sampling distribution of ^p N − ^p K ? Why?
b. Find the mean of the sampling distribution. Show work.
c. Find the standard deviation of the sampling distribution. Show work.
Confidence Intervals for p1-p2
Estimate Two-sample z interval for p1-p2 (2-PropZInt) Random: The data are producd by
a random sample of size n1 from
An approximate level C confidence interval for p1-p2 population 1 and a random sample
is of size n2 from population 2 or by
two groups of size n1 and n2 in a
√ ^p 1 ( 1−^p 1 ) ^p2 ( 1−^p2 ) randomized experiment.
( ^p1− ^p2 ) =± z ¿ +
n1 n2 Normal: The counts of “successes”
and “failures” in each sample or
where z* is the standard Normal critical value. group -- n1p1, n1(1-p1), n2p2, n2(1-p2)
– are at least 10.
Independent: Both the samples or
groups themselves and the
individual observations in each
sample or group are independent.
When sampling without
replacement, check that the two
populations are at least 10 times as
large as the corresponding samples
(the 10% condition).
Background - Often times we want to compare proportions of individuals with a certain characteristic
from two different populations. This might be the result of sampling two distinct populations or
comparing the effectiveness of two distinct treatments in an experiment.
The Sampling Distribution of a Difference between Two Proportions
From Chapter 7, we saw that the sampling distribution of ^p has the following properties:
Shape:
Center:
Spread:
We can use the formulas for combining two independent random variables to describe the sampling
distribution of ^p1− ^p2 :
Mean:
Standard Deviation:
The Sampling Distribution of ^p1− ^p2
Choose an SRS of size n1 from Population 1 with proportion of successes p1 and an independent SRS
of size n2 from Population 2 with proportion of successes p2.
Shape:
Center:
Spread:
, Example – Nathan and Kyle both work for the Department of Motor Vehicles (DMV), but in different
states. In Nathan’s state, 80% of the registered cars are made by American manufacturers. In Kyle’s
state, only 60% of the registered cars are made by American manufacturers. Nathan selects a random
sample of 100 cars from his state and Kyle selects a random sample of 70 cars from his state. Let
^p N − ^p K be the difference in the sample proportion of cars made by American manufacturers.
a. What is the shape of the sampling distribution of ^p N − ^p K ? Why?
b. Find the mean of the sampling distribution. Show work.
c. Find the standard deviation of the sampling distribution. Show work.
Confidence Intervals for p1-p2
Estimate Two-sample z interval for p1-p2 (2-PropZInt) Random: The data are producd by
a random sample of size n1 from
An approximate level C confidence interval for p1-p2 population 1 and a random sample
is of size n2 from population 2 or by
two groups of size n1 and n2 in a
√ ^p 1 ( 1−^p 1 ) ^p2 ( 1−^p2 ) randomized experiment.
( ^p1− ^p2 ) =± z ¿ +
n1 n2 Normal: The counts of “successes”
and “failures” in each sample or
where z* is the standard Normal critical value. group -- n1p1, n1(1-p1), n2p2, n2(1-p2)
– are at least 10.
Independent: Both the samples or
groups themselves and the
individual observations in each
sample or group are independent.
When sampling without
replacement, check that the two
populations are at least 10 times as
large as the corresponding samples
(the 10% condition).
