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Comparing Two Population
Means
Chapter 9
Introduction
What information does the test statistics give us?
What does a p-value mean in terms of the null
hypothesis?
Suppose we are investigating the GPA of USU
students. We collect a sample and find the
average GPA for the sample. Testing the null
hypothesis H0=2.7, we get a p-value p=0.001. Is
it safe to say that we have proven that the
average GPA is not 2.7? Explain.
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Introduction
A farmer would like to know which of two
brands of fertilizer results in the greater
average yield from his tomato plants.
How would you design a study to address
this question?
How would you analyze your results?
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Introduction
Comparing a treatment group to a control group
or one treatment to another to determine if the
mean response differs is an important tool of
research in many disciplines.
When comparing the means of two groups, a
researcher has two sets of data observations.
This is referred to as a two-sample problem.
Two-sample problems can involve either paired
(related) samples or independent (unrelated)
samples.
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Paired Samples
Paired samples arise when measures are made twice on the same
subject, or measures are made on two subjects that can be
considered to be dependent.
• To assess the effectiveness of a reading remediation course,
students are given a pre-test and a post-test and the average
scores are compared.
• The speed at which a group of athletes can run 1 mile is recorded,
they are then given a strict training regime for a period of time
and their mile times are recorded again.
• An experiment is conducted to compare a new tire to a standard
tire. One of each type of tire is placed on each of 20 trucks, the
trucks are driven over a variety of road conditions and the
reduction of tread is measured for each tire.
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Paired Samples
Paired samples are analyzed by reducing the problem to a
one sample problem. This is done by calculating the
differences between each of the pairs of observations.
We can apply the techniques of chapter 8 to make
inferences about the unknown mean μz of the
differences.
In general, for related samples we observe the n pairs (X1,
Y1), (X2, Y2),…, (Xn, Yn). The difference in the ith pair is
denoted by Zi = Xi – Yi, for i = 1, 2,…, n.
What’s a point estimate of μz? What is the standard error
of this point estimate?
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Paired Samples
We are usually interested in whether the mean μz of the
differences is equal to zero so we test the hypothesis
H0: μz = 0 versus HA: μz ≠ 0 using the test statistic
n ( z − 0)
t=
s
which will follow a t-distribution if the sample size is
sufficiently large and we assume the null hypothesis is
true.
Additionally, a confidence interval for μz is given by
tα / 2,n −1s
tα / 2,n −1s
⎛
μ Z = μ A − μ B ∈ ⎜⎜ z −
,z +
n
n
⎝
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⎞
⎟⎟
⎠
7
Paired Samples
Example: Neurobiology suggests that piano lessons may
improve the spatial-temporal reasoning of preschool
children. To test this hypothesis, the spatial-temporal
reasoning of 34 preschool children was measured
before and after 6 months of piano lessons. The
changes in their reasoning scores are shown below:
Is there evidence that
piano lessons changes
spatial-temporal
reasoning? Construct a
95% confidence interval
for the average
difference.
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Independent Samples
Independent samples arise when measures are made on two unrelated
or independent subjects.
• To assess the effectiveness of a reading remediation course,
students are randomly assigned to a remediation and nonremediation group. Reading scores are compared between the two
groups.
• Athletes are randomly assigned to two groups. The first is given a
rigorous new training regime and the second uses a standard
regime, the average mile times of the two groups are compared.
• New tires are put on 10 trucks and standard tires are put on 10
others. The trucks then drive over a variety of terrains. The
reduction in tread is measured and compared for the two types of
tires.
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Independent Samples
Example: We’d like to assess the effect of piano lessons
on spatial-temporal reasoning by comparing the piano
lesson group to a control group that did not receive
piano lessons. The changes in scores for the treatment
and control group are:
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Independent Samples
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Independent Samples
Consider a sample of n observations xi from
population A, with mean x and sample standard
deviation sx, and a sample of m observations yi
from population B with mean y and standard
deviation sy. The point estimate of the
difference in means is x − y, thus
s.e.( x − y ) =
σ A2
n
+
σ B2
m
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Independent Samples
There are two methods for analyzing independent samples - the
difference between them is how the standard error is estimated.
The book describes a third method for when the population
variances are known, but we will not discuss this, since
population variances are usually unknown.
The two procedures for estimating the variance are a “general
procedure” that can be used in any case and a “pooled variance
procedure” that is useful when the variances of the two
populations are approximately equal.
Once we have an estimate of the standard error of x − y , we
can use this estimate to create confidence intervals and conduct
hypothesis tests for the difference.
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Independent Samples
The general procedure estimates the standard
error
s.e.( x − y ) =
σ
2
A
n
+
σ
2
B
m
by s.e.( x − y ) =
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x
s y2
s
+
n m
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Independent Samples
Thus, a (1-α) level confidence interval for μA – μB is given
by
⎛
μ A − μ B ∈ ⎜ x − y − tα / 2,ν
⎜
⎝
2
x
s y2
s
+ , x − y + tα / 2,ν
n m
2 ⎞
s
s
y
+ ⎟
n m⎟
⎠
2
x
The degrees of freedom ν can be calculated via a formula
given of page 395 in your text. Statistical software will
calculate this automatically. When making calculations
by hand, we will use the convention that the degrees of
freedom ν are equal to the minimum of (n-1) and (m1).
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Independent Samples
To implement a hypothesis test for
H0: μA - μB = δ, the test statistic is
T=
x − y −δ
2
x
2
y
s
s
+
n m
Under the null hypothesis and given a sufficiently
large sample size, T approximately follows a tν
distribution, where ν = min(n-1,m-1).
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Independent Samples
We’d like to assess the effect of piano lessons on spatialtemporal reasoning by comparing the piano lesson
group to a control group that did not receive piano
lessons. The sample mean and standard deviation of
score changes for the treatment group are 3.62 and
3.06 respectively (n=34), while the mean and standard
deviation for the second sample are 0.39 and 2.42
(m=44).
Construct a 95% confidence interval for the difference in
score changes between the treatment and control
groups.
Conduct a hypothesis test to determine whether there is a
difference between the change in scores for the two
groups.
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Independent Samples
The general procedure can always be used if the sample
size is sufficiently large or the data are approximately
normally distributed. However, when it is safe to
assume that the variances of the two populations are
equal, a better analysis can be obtained by using the
estimate
s.e.( x − y ) = s p
where
sp =
1 1
+
n m
(n − 1) s x2 + (m − 1) s y2
n+m−2
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Independent Samples
A confidence interval for μA – μB is given by
⎛
1 1
1 1⎞
⎜
μ A − μ B ∈ ⎜ x − y − tα / 2,n + m − 2 s p
+ , x − y + tα / 2,n + m − 2 s p
+ ⎟⎟
n m
n m⎠
⎝
And a hypothesis test of H0: μA - μB = δ is implemented
using the test statistic
x − y −δ
T=
1 1
+
sp
n m
where T has a t-distribution with n+m-2 degrees of
freedom.
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Independent Samples
How do we decide if the sample variances are
approximately equal? A good rule of thumb is to
assume equal variances when the larger of s2x and s2y is
no more than 1.5 times the smaller of the two.
Returning to the ‘piano lessons affect spatial-temporal
reasoning’ example, is it appropriate to assume that
the variances are equal?
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Independent Samples
To study the question “does cocaine use by pregnant women
cause their babies to have low birth weight,” birth weights
(in lbs) of babies whose mothers used cocaine during
pregnancy were compared to birth weights of babies whose
mothers did not. For the groups whose mothers did use
cocaine, the average and standard deviation of birth weight
are 6.025 and 1.32 respectively. For the other group, the
average birth weight was 6.87 lbs with and standard
deviation of 1.48 pounds.
Is it appropriate to assume equal variances of the two samples?
Construct a 95% confidence interval for the difference in birth
weight between the ‘cocaine’ and ‘no cocaine’ groups.
Conduct a hypothesis test to determine whether there is a
difference in birth weight for the two groups.
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Independent Samples
A study looked at the relationship between physical fitness
and ego. Middle-aged college faculty were divided into
low and high fitness groups based on a physical exam and
were given a personality test to assess ego strength. The
average and standard deviation of ego strength scores for
the low fitness group were 4.64 and 0.69 respectively
(n=14), while the mean and standard deviation of the high
fitness group were 6.42 and 0.44 respectively (m=14).
Construct a 99% confidence interval for the difference in ego
scores of the two groups. Use the pooled variance
procedure if this is appropriate.
Is there evidence that mean ego scores are different in the
two groups?
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Independent Samples
Describe a two-sample experiment that would
be relevant to your field. How would you
collect you samples? Would you use paired or
independent samples?
Find a general form for a two-sided confidence
interval for a statistic θˆ.
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Statistical Software
In R two sample tests are implemented with the command:
>t.test(x,y,paired=F,var.equal=T)
“paired=F” and “var.equal=F” are the defaults.
Notes: Use the command x<-read.table(“file.txt”, header=T) to
read in files consisting of multiple columns.
Use the command x[i,j] to access row i and column j of x.
‘x[i,]’ will access row i and ‘x[,j]’ will access column j.
So if the two samples are in columns 1 and 2 of x, the command
to implement a two sample test for independent samples
would look like this (assuming unequal variances):
>t.test(x[,1],x[,2])
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Statistical Software
Excel “ttest(array1,array2,tails,type)”
– array1 and array2 contain the x and y samples
– ‘tails’ allows you to choose a two-sided or one-sided test
– ‘type’ options are ‘1-paired, 2-equal var, 3-unequal var’
SAS From ‘hypothesis tests’ under statistics menu
– ‘two sample paired t-test for means’: Enter variables into
groups 1 and 2, choose hypotheses, ok.
– ‘two sample t-test for means’: select ‘two variables’, enter
variables into groups 1 and 2, choose hypotheses, ok. This will
give you results for equal and unequal variances.
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Examples
State whether you would treat the samples as paired or
independent. If independent, would you use the
general or equal variance procedure to analyze the
data?
1. Does calcium reduce blood pressure? A randomized
comparative experiment gave one group of 10 men a
calcium supplement for 12 weeks. The control group of
11 men received a placebo. The average decrease in
blood pressure for the treatment group was 5 mm with
sample standard deviation 8.743 mm. The average and
standard deviation for the control (placebo) group were
-0.273 mm and 5.901 mm respectively. Is there
evidence of a difference between the two groups?
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Examples
State whether you would treat the samples as paired or
independent. If independent, would you use the general
or equal variance procedure to analyze the data?
2. A reading activity is introduced to help third graders
improve their reading. A class of 21 students takes part
in the activities for 8 weeks. A control class of 23
students has the same curriculum minus the activities.
At the end of the 8 weeks, all students are given a
reading test. The average and standard deviation of the
treatment groups scores are 51.41 and 11.01
respectively. The average and standard deviation of the
control group scores are 41.52 and 17.15 respectively.
Is there evidence that the activities helped?
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Examples
State whether you would treat the samples as paired or
independent. If independent, would you use the
general or equal variance procedure to analyze the
data?
3. A group of 133 male students and a group of 162
female students were given a test to determine how
accurately they appraise one another. The possible
test scores were from 0 to 41. The average scores and
standard deviations for the two groups were 25.24 and
5.05 for the males and 24.94 and 5.44 for the females.
Do these data support the contention that male and
female students differ in average social insight?
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Examples
State whether you would treat the samples as paired or
independent. If independent, would you use the
general or equal variance procedure to analyze the
data?
4. A test was given to the husband and wife in 133
married couples to determine how well men appraise
women and vice versa. The possible test scores were
from 0 to 41. The average scores and standard
deviations for the two groups were 25.24 and 5.05 for
the males and 24.94 and 5.44 for the females. Do
these data support the contention that males and
females differ in average social insight?
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Examples
State whether you would treat the samples as paired or
independent. If independent, would you use the general or
equal variance procedure to analyze the data?
5. A pharmaceutical company interested in comparing a new
drug for arthritis, drug A, to the standard treatment, drug B,
administers each of the drugs to a group of 30 arthritis
sufferers and records the time it takes for the pain to go
away. The order in which of the subjects receives the
treatments is random. The drugs are administers on
different days so that there are no residual effects from a
previous treatment. The average time for drug A was 15.2
minutes with a standard deviation of 3 minutes. The
average time for drug B was 17.3 minutes with a standard
deviation of 5 minutes. Is there evidence that there is a
difference in relief times?
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