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Chapter 24
Comparing Means
Copyright © 2009 Pearson Education, Inc.
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Slide
Slide241- 3
Main topics of this chapter




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In this chapter, we study the sampling difference
for the difference between two means.
Two-sample t-test
Two-sample t-interval
Assumptions and conditions
Equality of variances
Copyright © 2009 Pearson Education, Inc.
Slide
Slide241- 4
Division of Mathematics, HCC
Course Objectives for Chapter 24
After studying this chapter, the student will be able to:
64.
Perform a two-sample t-test for two population means, to
include:

writing appropriate hypotheses,

checking the necessary assumptions and conditions,

drawing an appropriate diagram,

computing the P-value,

making a decision, and

Interpreting the results in the context of the problem.
65.
Find a t-based confidence interval for the difference
between two population means, to include:

Finding the t that corresponds to the percent

Computing the interval

Interpreting the interval in context.
Copyright © 2009 Pearson Education, Inc.
Let’s start with an example




A student was curious about whether generic or
brand-named batteries performed better in a CD
player.
He randomly selected six of each type of battery.
NOTE: He went to different stores since batteries
that are shipped together have more of a
tendency to fail together.
He tested them on the same CD player to see
how long they would run with continuous play.
Copyright © 2009 Pearson Education, Inc.
Slide
Slide241- 6
Let’s start with an example


The six generic batteries
ran for an average of 206
minutes with a standard
deviation of 10.3 minutes.
The six name-brand
batteries ran for an
average of 187.4 minutes
and a standard deviation
of 14.6 minutes.
Brand Name
Generic
194.0
190.7
205.5
203.5
199.2
203.5
172.4
206.5
184.0
222.5
169.5
209.4
Note: Data are included in the 3rd
edition, but not the first two.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 7
Copyright © 2009 Pearson Education, Inc.
Slide 1- 8
The W’s

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Who: Name brand and generic batteries
What: Difference in Battery Life
Why: To estimate the true mean difference in the
battery life of generic and name-brand batteries
When: Recently (doesn’t say)
Where: Doesn’t say
How: Purchased in stores around town.
Copyright © 2009 Pearson Education, Inc.
Slide
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Plot the Data


The natural display for
comparing two groups is
boxplots of the data for the
two groups, placed sideby-side. For example:
Recall: We did this in
Chapter 5.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 10
Recall – Comparing two boxplots
Copyright © 2009 Pearson Education, Inc.
Slide 1- 11
Copyright © 2009 Pearson Education, Inc.
Slide 1- 12
Copyright © 2009 Pearson Education, Inc.
Slide 1- 13
Comparing Two Means



Once we have examined the side-by-side
boxplots, we can turn to the comparison of two
means.
Comparing two means is not very different from
comparing two proportions.
This time the parameter of interest is the
difference between the two means, 1 – 2.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 14
Comparing Two Means (cont.)


Remember that, for independent random
quantities, variances add.
So, the standard deviation of the difference
between two sample means is
SD  y1  y2  

 12
n1

 22
n2
We still don’t know the true standard deviations of
the two groups, so we need to estimate and use
the standard error
s12 s22
SE  y1  y2  

n1 n2
Copyright © 2009 Pearson Education, Inc.
Slide 1- 15
Comparing Two Means (cont.)

Because we are working with means and
estimating the standard error of their difference
using the data, we shouldn’t be surprised that the
sampling model is a Student’s t.
 The confidence interval we build is called a
two-sample t-interval (for the difference in
means).
 The corresponding hypothesis test is called a
two-sample t-test.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 16
Sampling Distribution for the Difference
Between Two Means

When the conditions are met, the standardized sample
difference between the means of two independent groups
y1  y2    1  2 

t
SE  y1  y2 

can be modeled by a Student’s t-model with a number of
degrees of freedom found with a special formula.
We estimate the standard error with
s12 s22
SE  y1  y2  

n1 n2
Copyright © 2009 Pearson Education, Inc.
Slide 1- 17
Assumptions and Conditions

Independence Assumption (Each condition needs to be
checked for both groups.):
 Randomization Condition: Were the data collected with
suitable randomization (representative random
samples or a randomized experiment)?
 10% Condition: We don’t usually check this condition
for differences of means. We will check it for means
only if we have a very small population or an extremely
large sample. However, my view is that this condition
should be checked. Otherwise, what do we mean by a
“small population” or a “large sample”? I think that the
authors are saying that this condition is either
obviously satisfied or obviously not satisfied!
Copyright © 2009 Pearson Education, Inc.
Slide 1- 18
Assumptions and Conditions (cont.)


Normal Population Assumption:
 Nearly Normal Condition: This must be checked for
both groups. A violation by either one violates the
condition.
Independent Groups Assumption: The two groups we are
comparing must be independent of each other. (See
Chapter 25 if the groups are not independent of one
another…)
Copyright © 2009 Pearson Education, Inc.
Slide 1- 19
For the battery example

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Randomization: Batteries were selected
randomly. Hopefully the student sampled from
different cartons – preferably even at different
stores. We’ll assume that this is so.
Independent Groups: The batteries were made
by two different companies.
Nearly Normal Condition: Hard to tell because of
small samples. We’ll give it to them for now.
10% condition: Obvious.
Copyright © 2009 Pearson Education, Inc.
Slide
Slide241- 20
Two-Sample t-Interval
When the conditions are met, we are ready to find the confidence
interval for the difference between means of two independent groups.
The confidence interval is
 y1  y2   t

df
 SE  y1  y2 
where the standard error of the difference of the means is
s12 s22
SE  y1  y2  

n1 n2
The critical value depends on the particular confidence level, C, that you
specify and on the number of degrees of freedom, which we get from the
sample sizes and a special formula.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 21
Degrees of Freedom

The special formula for the degrees of freedom
for our t critical value is a bear:
2
 s12 s22 
  
n1 n2 

df 
2
2
2
2




s1
s2
1
1

 
 
n1  1  n1  n2  1  n2 

Because of this, we will let technology calculate
degrees of freedom for us!
Copyright © 2009 Pearson Education, Inc.
Slide 1- 22
Copyright © 2009 Pearson Education, Inc.
Slide 1- 23
Just for the record --


The “bear” is actually called the Satterwaithe
approximation to the degrees of freedom.
The “Satterwaithe approximation”, rather than
the “bear”, is the accepted statistical terminology
But “bear” is easier to spell than Satterwaithe!
Copyright © 2009 Pearson Education, Inc.
Slide
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WOW!!!
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The problem is that the variances are not the same.
We have to make some assumption about what the
combined variances will look like.
There are two ways out of this.
One way is to assume that the variances are the same
(and sometimes we can.)
The other is to “pool” variances.
Or we can “hug the bear”!
We’ll come back to this. But let’s pursue our example
and hug the bear for now.
Copyright © 2009 Pearson Education, Inc.
Slide
Slide241- 25
For our battery example


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You can plug the numbers into the formulas on
the last slides - including the bear!
But there is a lot of room for miscalculation –
 Forgetting to square the variances in the SE
 Forgetting to take the square root in the SE
 Doing a number of things in the “bear”
 And who knows what else!
Take my word for it!
Let technology do the work on this one!
Copyright © 2009 Pearson Education, Inc.
Slide
Slide241- 26
Slight problem with StatCrunch


We cannot do the exercise using the worksheet
that the publisher provided.
I retyped the data as shown in the next slide.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 27
Copyright © 2009 Pearson Education, Inc.
Slide 1- 28
Testing the Difference Between Two Means


The hypothesis test we use is the two-sample ttest for the difference between means.
The conditions for the two-sample t-test for the
difference between the means of two
independent groups are the same as for the twosample t-interval.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 29
A Test for the Difference Between Two Means
We test the hypothesis H0: 1 – 2 = 0, where the
hypothesized difference, 0, is almost always 0, using the
statistic
y1  y2    0

t
SE  y1  y2 
The standard error is
s12 s22
SE  y1  y2  

n1 n2
When the conditions are met and the null hypothesis is
true, this statistic can be closely modeled by a Student’s
t-model with a number of degrees of freedom given by a
special formula. We use that model to obtain a P-value.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 30
For our battery example – StatCrunch

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Select Stat, then , then t-statistics, then 2sample, then “with data”.
We have the data. However, if you have
summary statistics, enter them as requested –
don’t mix anything up!
Select
 Confidence Interval (95%) or
 Hypothesis Test, 0 and ≠
Then Calculate.
Copyright © 2009 Pearson Education, Inc.
Slide
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Copyright © 2009 Pearson Education, Inc.
Slide 1- 32
Intermediate steps (without pooling)
Copyright © 2009 Pearson Education, Inc.
Slide 1- 33
Confidence Interval
Copyright © 2009 Pearson Education, Inc.
Hypothesis Test
Slide 1- 34
95% confidence interval results:
μ1 : mean of Brand Name
μ2 : mean of Generic
μ1 - μ2 : mean difference
(without pooled variances)
Difference
μ1 - μ2
Sample
Mean
-18.583334
Copyright © 2009 Pearson Education, Inc.
Std. Err.
7.298451
DF
8.9862795
L. Limit
-35.09742
U. Limit
-2.0692465
Slide 1- 35
Hypothesis test results:
μ1 : mean of Brand Name
μ2 : mean of Generic
μ1 - μ2 : mean difference
H0 : μ1 - μ2 = 0
HA : μ1 - μ2 ≠ 0
(without pooled variances)
Difference
μ1 - μ2
Sample
Mean
-18.583334
Copyright © 2009 Pearson Education, Inc.
Std. Err.
7.298451
DF
8.9862795
T-Stat
-2.5462024
P-value
0.0314
Slide 1- 36
For our battery example - TI

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There is minor
disagreement in the
answer with StatCrunch
Press STAT, tab to Tests,
then (on the 84) go to (0)
– 2SampTInt
Select Stats.
Then enter the summary
statistics as requested –
again, don’t mix anything
up (it’s not that hard to
mix stuff up!)
Copyright © 2009 Pearson Education, Inc.
Slide
Slide241- 37
For our battery example - TI


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Select C-level .95, and
say NO to Pool.
The TI gives (2.0958,
35.104).
StatCrunch gave
(2.069, 35.097).
Very slight difference,
most likely rounding
and slight difference in
df (again, rounding).
Both used the bear!
Copyright © 2009 Pearson Education, Inc.
The Bear:
TI: 8.988928812
SC: 8.986279
Slide 1- 38
Battery example – with the TI (84+)




Start over:
Select STAT, tab to Tests,
then select “2SampTtest”.
Input data as before (it may
already be there from your
prior calculation)
Pooled – NO; Alternative:
Unequal, then Calculate
Copyright © 2009 Pearson Education, Inc.
Slide
Slide241- 39
Results of pooling.

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Degrees of freedom are different (10, or the sum
of the sample sizes minus two).
Confidence Interval is (6.83, 25.1)
Hypothesis test: p –value = 0.029.
See next two slides.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 40
95% confidence interval results:
μ1 : mean of Times
μ2 : mean of Brand Name
μ1 - μ2 : mean difference
(with pooled variances)
Difference
μ1 - μ2
Sample
Mean
9.291667
Copyright © 2009 Pearson Education, Inc.
Std. Err.
7.604908
DF
L. Limit
16
-6.8300176
U. Limit
25.413351
Slide 1- 41
Hypothesis test results:
μ1 : mean of Brand Name
μ2 : mean of Generic
μ1 - μ2 : mean difference
H0 : μ1 - μ2 = 0
HA : μ1 - μ2 ≠ 0
(without pooled variances)
Difference
μ1 - μ2
Sample
Mean
-18.583334
Copyright © 2009 Pearson Education, Inc.
Std. Err.
7.298451
DF
8.98627
95
T-Stat
-2.5462024
P-value
0.0314
Slide 1- 42
Hypothesis test results:
μ1 : mean of Brand Name
μ2 : mean of Generic
μ1 - μ2 : mean difference
H0 : μ1 - μ2 = 0
HA : μ1 - μ2 ≠ 0
(with pooled variances)
Difference
μ1 - μ2
Sample
Mean
-18.583334
Copyright © 2009 Pearson Education, Inc.
Std. Err.
7.298451
DF
T-Stat
10 -2.5462024
P-value
0.0291
Slide 1- 43
Copyright © 2009 Pearson Education, Inc.
Slide 1- 44
Battery example – with the TI (84+)




Start over:
Select STAT, tab to Tests,
then select “2SampTtest”.
Input data as before (it may
already be there from your
prior calculation)
Pooled – NO; Alternative:
Unequal, then Calculate
Copyright © 2009 Pearson Education, Inc.
Slide
Slide241- 45
Conclusions in context
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
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
Important to note: The confidence interval does
not include 0.
This leads one to conclude (no pooling)
“We can say with 95% confidence that, based on
our sample of six batteries of each type, the
interval for the difference in mean life batteries is
from 2.1 to 35.1.“
“If the batteries had the same life, we would have
a result this extreme or more 3.1% (p = 0.031) of
the time. At α = 0.05, there is sufficient evidence
to reject the hypothesis of no difference in
battery life.”
Copyright © 2009 Pearson Education, Inc.
Slide
Slide241- 46
**2 sample t-test with EXCEL



One way : T.TEST(array1,array2,tails,pool)
 Pool = 2 if pooling
 Pool = 3 if no pooling (recommended)
 See next slide
Another way: Use the Data Analysis toolpak
 Select 2-sample t (unequal variances)
nd through 4th slides after this one
 See 2
EXCEL does not do the confidence interval.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 47
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Slide 1- 48
Copyright © 2009 Pearson Education, Inc.
Slide 1- 49
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Slide 1- 50
Copyright © 2009 Pearson Education, Inc.
Slide 1- 51
Back Into the Pool

Remember that when we know a proportion, we
know its standard deviation.
 Thus, when testing the null hypothesis that two
proportions were equal, we could assume their
variances were equal as well.
 This led us to pool our data for the hypothesis
test.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 52
Back Into the Pool (cont.)

For means, there is also a pooled t-test.
 Like the two-proportions z-test, this test
assumes that the variances in the two groups
are equal.
 But, be careful, there is no link between a
mean and its standard deviation…
Copyright © 2009 Pearson Education, Inc.
Slide 1- 53
Back Into the Pool (cont.)


If we are willing to assume that the variances of
two means are equal, we can pool the data from
two groups to estimate the common variance and
make the degrees of freedom formula much
simpler.
We are still estimating the pooled standard
deviation from the data, so we use Student’s
t-model, and the test is called a pooled t-test (for
the difference between means).
Copyright © 2009 Pearson Education, Inc.
Slide 1- 54
*The Pooled t-Test

If we assume that the variances are equal, we can
estimate the common variance from the numbers we
already have:
s

2
pooled
n1  1 s   n2  1 s


 n1  1   n2  1
2
1
Substituting into our standard error formula, we get:
SE pooled  y1  y2   s pooled

2
2
1 1

n1 n2
Our degrees of freedom are now df = n1 + n2 – 2.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 55
*The Pooled t-Test and Confidence Interval
for Means

The conditions for the pooled t-test and corresponding
confidence interval are the same as for our earlier twosample t procedures, with the assumption that the
variances of the two groups are the same.
For the hypothesis test, our test statistic is

which has df = n1 + n2 – 2.
Our confidence interval is

y1  y2    0

t
SE pooled  y1  y2 
 y1  y2   t
Copyright © 2009 Pearson Education, Inc.

df
 SE pooled  y1  y2 
Slide 1- 56
Is the Pool All Wet?



So, when should you use pooled-t methods
rather than two-sample t methods? Never. (Well,
hardly ever.)
Because the advantages of pooling are small,
and you are allowed to pool only rarely (when the
equal variance assumption is met), don’t.
It’s never wrong not to pool.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 57
Why Not Test the Assumption That the
Variances Are Equal?



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There is a hypothesis test that would do this.
But, it is very sensitive to failures of the
assumptions and works poorly for small sample
sizes—just the situation in which we might care
about a difference in the methods.
So, the test does not work when we would need it
to.
I’ll tell you what it is, though.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 58
*Testing for equal variances






Suppose you have two samples with summary statistics
(n1, ȳ1 ,s1) and (n2, ȳ2, s2). She 1 subscript is supposed
to be the larger variance, but the technology will
rearrange.
Both the TI and StatCrunch use the F-Test. (We use
another at FDA; “Levene’s Test”).
Compute the ratio of (s12/n1) to (s22/n2). (Here the n’s are
the same.)
We get p = 0.4621 with both the TI and StatCrunch;
p = 0.711 with MINITAB; they use a correction
(Bonferroni)
The variances are the same!
Copyright © 2009 Pearson Education, Inc.
Slide
Slide241- 59
*Testing for equal
variances (TI)



[STATS],Tests,(E) 2SampFTest
Enter data as shown –
group variances and
sample size.
See result


P = 0.4621
Variances are the
same.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 60
Is There Ever a Time When Assuming Equal
Variances Makes Sense?



Yes. In a randomized comparative experiment,
we start by assigning our experimental units to
treatments at random.
Each treatment group therefore begins with the
same population variance.
In this case assuming the variances are equal is
still an assumption, and there are conditions that
need to be checked, but at least it’s a plausible
assumption.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 61
What Can Go Wrong?


Watch out for paired data.
 The Independent Groups Assumption
deserves special attention.
 If the samples are not independent, you can’t
use two-sample methods.
 This is why we need Chapter 25!
Look at the plots.
 Check for outliers and non-normal distributions
by making and examining boxplots.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 62
What have we learned?

We’ve learned to use statistical inference to
compare the means of two independent groups.




We use t-models for the methods in this chapter.
It is still important to check conditions to see if our
assumptions are reasonable.
The standard error for the difference in sample means
depends on believing that our data come from
independent groups, but pooling is not the best choice
here.
The reasoning of statistical inference remains the
same; only the mechanics change.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 63
Main topics of this chapter





In this chapter, we study the sampling difference
for the difference between two means.
Two-sample t-test
Two-sample t-interval
Assumptions and conditions
Equality of Variances (thrown in!)
Copyright © 2009 Pearson Education, Inc.
Slide
Slide241- 64
Division of Mathematics, HCC
Course Objectives for Chapter 24
After studying this chapter, the student will be able to:
64.
Perform a two-sample t-test for two population means, to
include:

writing appropriate hypotheses,

checking the necessary assumptions and conditions,

drawing an appropriate diagram,

computing the P-value,

making a decision, and

Interpreting the results in the context of the problem.
65.
Find a t-based confidence interval for the difference
between two population means, to include:

Finding the t that corresponds to the percent

Computing the interval

Interpreting the interval in context.
Copyright © 2009 Pearson Education, Inc.