8.0: Comparisons Involving Means
and Proportions
Note: For access to statistical tables and equations, please click "Help Files" on the
upper right of any module/exam page.
So far, we have considered interval estimation and hypothesis testing for a single
population. In this module, we will study interval estimation and hypothesis testing for
differences between two population means as well as for differences between two
population proportions.
We will begin this study with two definitions:
A sampling method is called dependent if the items (or the individuals) that are selected
in one sample are used in order to determine the items (or individuals) in the second
sample.
For example, suppose you believe that women marry men who are about the same age
as themselves. In order to see if this is true, you select 100 married women and see if
their husbands are about the same age. This is an example of dependent
sampling because once you have selected each woman for your study, you automatically
get a specific man (her husband).
A sampling method is called independent if the items (or the individuals) that are
selected in one sample do not determine the items (or individuals) in the second sample.
For example, suppose you believe that fifth graders at Parker Elementary School read
better than fifth graders at Central Elementary School. You test your hypothesis by
giving a reading test to 50 fifth graders from Parker Elementary School and by giving the
same reading test to 65 fifth graders from Central Elementary School. This is an example
of independent sampling because the students you choose from Parker Elementary do
not determine the specific students that you select from Central Elementary.
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,8.1: Independent Samples:
Inferences Involving Differences
Between Means
Independent Samples: Inferences Involving Differences
Between Means
We often desire to find confidence intervals for the difference between two population
means. First of all, we will consider independent samples. Suppose that:
µ1 and µ2 are the means for population 1 and for population 2, respectively.
σ1 and σ2, are the standard deviations for population 1 and for population 2, respectively.
n1 and n2 are the number of samples from population 1 and from population 2,
respectively.
x1 and x2 are the means for sample 1 and sample 2, respectively.
s1 and s2, are the standard deviations for sample 1 and sample 2, respectively.
We find the confidence interval for difference of the two means (µ1 - µ2) by using:
(eqn. 8.1)
So, eqn 8.1 give us the confidence interval for the difference of the two population
means. In order to use eqn 8.1, the sample sizes from each of our populations must be
greater than or equal to 30, i.e., n1 ≥ 30 and n2 ≥ 30.
After using equation 8.1 to calculate the confidence interval, we interpret the results in
the following way. Please note, all of these explanations assume we are using a 90%
confidence interval; if the confidence interval is different, you would change the percent.
1. If the entire confidence interval is positive (it is never negative or zero), then we can
say that we are 90% confident that there is a difference in the two population means.
Since the entire confidence interval is positive, we can be 90% confident that µ1 minus
, µ2 is positive. This means we are 90% confident that the mean for population 1 is
greater than the mean for population 2.
2. If the entire confidence interval is negative (it is never positive or zero), then we can
say that we are 90% confident that there is a difference in the two population means.
Since the entire confidence interval is negative, we can be 90% confident that µ1 minus
µ2 is negative. This means we are 90% confident that the mean for population 2 is
greater than the mean for population 1.
3. If the confidence interval covers both positive and negative values, then we cannot be
90% confident that there is a difference in the two population means.
We will illustrate the use of eqn. 8.1 by a number of examples.
Example 8.1
Independent random samples were selected from population 1 and population 2. The
following information was obtained from these samples:
a) Find the 90% confidence interval for estimating the difference in the population
means (µ1 - µ2).
b) Can you be 90% confident that there is a difference in the means of the two
populations?
Solution
When we look back at table 6.1, we see that 90% confidence corresponds to z = 1.645.
a) Notice that the sample sizes are each greater than 30, so we may use eqn. 8.1:
and Proportions
Note: For access to statistical tables and equations, please click "Help Files" on the
upper right of any module/exam page.
So far, we have considered interval estimation and hypothesis testing for a single
population. In this module, we will study interval estimation and hypothesis testing for
differences between two population means as well as for differences between two
population proportions.
We will begin this study with two definitions:
A sampling method is called dependent if the items (or the individuals) that are selected
in one sample are used in order to determine the items (or individuals) in the second
sample.
For example, suppose you believe that women marry men who are about the same age
as themselves. In order to see if this is true, you select 100 married women and see if
their husbands are about the same age. This is an example of dependent
sampling because once you have selected each woman for your study, you automatically
get a specific man (her husband).
A sampling method is called independent if the items (or the individuals) that are
selected in one sample do not determine the items (or individuals) in the second sample.
For example, suppose you believe that fifth graders at Parker Elementary School read
better than fifth graders at Central Elementary School. You test your hypothesis by
giving a reading test to 50 fifth graders from Parker Elementary School and by giving the
same reading test to 65 fifth graders from Central Elementary School. This is an example
of independent sampling because the students you choose from Parker Elementary do
not determine the specific students that you select from Central Elementary.
Skip To Content
,8.1: Independent Samples:
Inferences Involving Differences
Between Means
Independent Samples: Inferences Involving Differences
Between Means
We often desire to find confidence intervals for the difference between two population
means. First of all, we will consider independent samples. Suppose that:
µ1 and µ2 are the means for population 1 and for population 2, respectively.
σ1 and σ2, are the standard deviations for population 1 and for population 2, respectively.
n1 and n2 are the number of samples from population 1 and from population 2,
respectively.
x1 and x2 are the means for sample 1 and sample 2, respectively.
s1 and s2, are the standard deviations for sample 1 and sample 2, respectively.
We find the confidence interval for difference of the two means (µ1 - µ2) by using:
(eqn. 8.1)
So, eqn 8.1 give us the confidence interval for the difference of the two population
means. In order to use eqn 8.1, the sample sizes from each of our populations must be
greater than or equal to 30, i.e., n1 ≥ 30 and n2 ≥ 30.
After using equation 8.1 to calculate the confidence interval, we interpret the results in
the following way. Please note, all of these explanations assume we are using a 90%
confidence interval; if the confidence interval is different, you would change the percent.
1. If the entire confidence interval is positive (it is never negative or zero), then we can
say that we are 90% confident that there is a difference in the two population means.
Since the entire confidence interval is positive, we can be 90% confident that µ1 minus
, µ2 is positive. This means we are 90% confident that the mean for population 1 is
greater than the mean for population 2.
2. If the entire confidence interval is negative (it is never positive or zero), then we can
say that we are 90% confident that there is a difference in the two population means.
Since the entire confidence interval is negative, we can be 90% confident that µ1 minus
µ2 is negative. This means we are 90% confident that the mean for population 2 is
greater than the mean for population 1.
3. If the confidence interval covers both positive and negative values, then we cannot be
90% confident that there is a difference in the two population means.
We will illustrate the use of eqn. 8.1 by a number of examples.
Example 8.1
Independent random samples were selected from population 1 and population 2. The
following information was obtained from these samples:
a) Find the 90% confidence interval for estimating the difference in the population
means (µ1 - µ2).
b) Can you be 90% confident that there is a difference in the means of the two
populations?
Solution
When we look back at table 6.1, we see that 90% confidence corresponds to z = 1.645.
a) Notice that the sample sizes are each greater than 30, so we may use eqn. 8.1:
