ST 305 Applied Statistics for Health Sciences
Hypothesis Testing – Compare Means of Two
Populations
Bin Wang
bwang@southalabama.edu
Department of Mathematics and Statistics
University of South Alabama
Spring 2014
Last updated on March 19, 2014
1/7
Hypothesis Testing for Two Population Means
An Example: Chocolate and Memory
Example (Chocolate and Memory)
Does eating chocolate improve memory?
Nine (9) people were given a memory test before and after eating
chocolate. The data for the number of words recalled out of 50 are shown
below. Assume Normality.
Before
After
Spring 2014
24
26
16
20
33
29
9
11
42
42
38
39
Last updated on March 19, 2014
27
25
30
34
41
44
2/7
Hypothesis Testing for Two Population Means
An Example: Chocolate and Memory
Example (Chocolate and Memory)
Does eating chocolate improve memory?
Nine (9) people were given a memory test before and after eating
chocolate. The data for the number of words recalled out of 50 are shown
below. Assume Normality.
Before
After
24
26
16
20
33
29
9
11
42
42
38
39
27
25
30
34
41
44
How many populations are there?
Two populations: all people before eating chocolate, and all people
after eating chocolate.
Spring 2014
Last updated on March 19, 2014
2/7
Hypothesis Testing for Two Population Means
An Example: Chocolate and Memory
Example (Chocolate and Memory)
Does eating chocolate improve memory?
Nine (9) people were given a memory test before and after eating
chocolate. The data for the number of words recalled out of 50 are shown
below. Assume Normality.
Before
After
24
26
16
20
33
29
9
11
42
42
38
39
27
25
30
34
41
44
How many populations are there?
Two populations: all people before eating chocolate, and all people
after eating chocolate.
Two samples: nine people before eating chocolate, and the same
group of nine people after eating chocolate.
Spring 2014
Last updated on March 19, 2014
2/7
Hypothesis Testing for Two Population Means
An Example: Chocolate and Memory
Example (Chocolate and Memory)
Does eating chocolate improve memory?
Nine (9) people were given a memory test before and after eating
chocolate. The data for the number of words recalled out of 50 are shown
below. Assume Normality.
Before
After
24
26
16
20
33
29
9
11
42
42
38
39
27
25
30
34
41
44
How many populations are there?
Two populations: all people before eating chocolate, and all people
after eating chocolate.
Two samples: nine people before eating chocolate, and the same
group of nine people after eating chocolate.
Are the two samples independent?
Spring 2014
Last updated on March 19, 2014
2/7
Hypothesis Testing for Two Population Means
An Example: Chocolate and Memory
Example (Chocolate and Memory)
Does eating chocolate improve memory?
Nine (9) people were given a memory test before and after eating
chocolate. The data for the number of words recalled out of 50 are shown
below. Assume Normality.
Before
After
24
26
16
20
33
29
9
11
42
42
38
39
27
25
30
34
41
44
What are the parameters being interested?
The means of the two populations:
the average words recalled by all people before eating chocolate,
denoted as µbefore ;
the average words recalled by all people after eating chocolate,
denoted as µafter ;
Spring 2014
Last updated on March 19, 2014
2/7
Hypothesis Testing for Two Population Means
An Example: Chocolate and Memory
Example (Chocolate and Memory)
Does eating chocolate improve memory?
Nine (9) people were given a memory test before and after eating
chocolate. The data for the number of words recalled out of 50 are shown
below. Assume Normality.
Before
After
24
26
16
20
33
29
9
11
42
42
38
39
27
25
30
34
41
44
What is the hypothesis we want to test?
If answer is “yes”, µbefore < µafter or equivalently µbefore − µafter < 0.
If answer is “no”, µbefore ≥ µafter or equivalently µbefore − µafter ≥ 0.
We can state the null and alternative hypotheses as follows:
H0 : µbefore − µafter = 0
Spring 2014
Ha : µbefore − µafter < 0
Last updated on March 19, 2014
2/7
Hypothesis Testing for Two Population Means
An Example: Chocolate and Memory
Paired t-test
In this example, the two samples are paired. We use a paired t-test to
analyze the data.
Spring 2014
Last updated on March 19, 2014
3/7
Hypothesis Testing for Two Population Means
An Example: Chocolate and Memory
Paired t-test
In this example, the two samples are paired. We use a paired t-test to
analyze the data.
The main idea is to compute the difference between each pair of measures
Di = Xi,before − Xi,after
Spring 2014
Last updated on March 19, 2014
3/7
Hypothesis Testing for Two Population Means
An Example: Chocolate and Memory
Paired t-test
In this example, the two samples are paired. We use a paired t-test to
analyze the data.
The main idea is to compute the difference between each pair of measures
Di = Xi,before − Xi,after
If we setup H0 and Ha differently, we need to compute the difference in
other ways. For example, if we have
H0 : µafter − µbefore = 0
Ha : µafter − µbefore > 0
Di = Xi,after − Xi,before
Spring 2014
Last updated on March 19, 2014
3/7
Hypothesis Testing for Two Population Means
An Example: Chocolate and Memory
Paired t-test
To apply the paired t-test, the sample size needs to be large (to use
the CLT).
Spring 2014
Last updated on March 19, 2014
4/7
Hypothesis Testing for Two Population Means
An Example: Chocolate and Memory
Paired t-test
To apply the paired t-test, the sample size needs to be large (to use
the CLT).
In practice, the sample size is often small in matched pairs design. If
so, we need to assume the differences D follows approximately
Normal distribution to use the paired t-test. Otherwise, we can
consider non-parametric test.
Spring 2014
Last updated on March 19, 2014
4/7
Hypothesis Testing for Two Population Means
An Example: Chocolate and Memory
Paired t-test
Spring 2014
To apply the paired t-test, the sample size needs to be large (to use
the CLT).
In practice, the sample size is often small in matched pairs design. If
so, we need to assume the differences D follows approximately
Normal distribution to use the paired t-test. Otherwise, we can
consider non-parametric test.
Performed paired t-test using StatCrunch (demonstration).
Last updated on March 19, 2014
4/7
Hypothesis Testing for Two Population Means
An Example: Chocolate and Memory
Paired t-test
Find the p-value, and make a decision:
If the p-value is smaller than the significance level, we reject H0 ,
otherwise, we don’t reject H0 . In this problem, we don’t reject H0 .
H0 : µbefore − µafter = 0
Spring 2014
Ha : µbefore − µafter < 0
Last updated on March 19, 2014
5/7
Hypothesis Testing for Two Population Means
An Example: Chocolate and Memory
Paired t-test
Find the p-value, and make a decision:
If the p-value is smaller than the significance level, we reject H0 ,
otherwise, we don’t reject H0 . In this problem, we don’t reject H0 .
H0 : µbefore − µafter = 0
Ha : µbefore − µafter < 0
In other words, we don’t have enough evidence to conclude H0 is
false. The evidence from the experiment doesn’t show that eating
chocolate will improve memory.
Spring 2014
Last updated on March 19, 2014
5/7
Hypothesis Testing for Two Population Means
Two independent samples t-test
Example (Chocolate and Memory (revised))
Does eating chocolate improve memory?
Nine (9) people were given a memory test without eating chocolate, and
another group of nine people were given a memory test after eating
chocolate. The data for the number of words recalled out of 50 are shown
below. Assume Normality.
Spring 2014
Without Chocolate
With Chocolate
24
26
16
20
33
29
9
11
42
42
Last updated on March 19, 2014
38
39
27
25
30
34
41
44
6/7
Hypothesis Testing for Two Population Means
Two independent samples t-test
Example (Chocolate and Memory (revised))
Does eating chocolate improve memory?
Nine (9) people were given a memory test without eating chocolate, and
another group of nine people were given a memory test after eating
chocolate. The data for the number of words recalled out of 50 are shown
below. Assume Normality.
Without Chocolate
With Chocolate
24
26
16
20
33
29
9
11
42
42
38
39
27
25
30
34
41
44
The two samples that are independent, not paired.
Spring 2014
Last updated on March 19, 2014
6/7
Hypothesis Testing for Two Population Means
Two independent samples t-test
Example (Chocolate and Memory (revised))
Does eating chocolate improve memory?
Nine (9) people were given a memory test without eating chocolate, and
another group of nine people were given a memory test after eating
chocolate. The data for the number of words recalled out of 50 are shown
below. Assume Normality.
Without Chocolate
With Chocolate
24
26
16
20
33
29
9
11
42
42
38
39
27
25
30
34
41
44
The two samples that are independent, not paired.
The subjects involved in the study was doubled.
Spring 2014
Last updated on March 19, 2014
6/7
Hypothesis Testing for Two Population Means
Two independent samples t-test
Example (Chocolate and Memory (revised))
Does eating chocolate improve memory?
Nine (9) people were given a memory test without eating chocolate, and
another group of nine people were given a memory test after eating
chocolate. The data for the number of words recalled out of 50 are shown
below. Assume Normality.
Without Chocolate
With Chocolate
24
26
16
20
33
29
9
11
42
42
38
39
27
25
30
34
41
44
The two samples that are independent, not paired.
The subjects involved in the study was doubled.
Individual effects.
Spring 2014
Last updated on March 19, 2014
6/7
Hypothesis Testing for Two Population Means
Two independent samples t-test
Independent samples t-test with StatCrunch
Spring 2014
Check assumptions: large sample/normality.
Last updated on March 19, 2014
7/7
Hypothesis Testing for Two Population Means
Two independent samples t-test
Independent samples t-test with StatCrunch
Spring 2014
Check assumptions: large sample/normality.
Population standard deviations known?
Last updated on March 19, 2014
7/7
Hypothesis Testing for Two Population Means
Two independent samples t-test
Independent samples t-test with StatCrunch
Spring 2014
Check assumptions: large sample/normality.
Population standard deviations known?
Are the two population standard deviations equal?
Last updated on March 19, 2014
7/7
Hypothesis Testing for Two Population Means
Two independent samples t-test
Independent samples t-test with StatCrunch
Spring 2014
Check assumptions: large sample/normality.
Population standard deviations known?
Are the two population standard deviations equal?
StatCrunch demonstration.
Last updated on March 19, 2014
7/7