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Hypothesis Testing
Comparisons Among Two Samples
Hypothesis Testing w/Two
Samples
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By far a more common statistic to use
than any other covered so far
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If we have access to an entire population
to calculate its mean (μ), why do we need
to get a sample to infer its characteristics?
We can just measure them directly
99% of the time we don’t have this kind of
access
Hypothesis Testing w/Two
Samples
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Also, most experiment employ two groups
– a treatment group and a control group
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The Treatment Group – gets the IV
The Control Group – identical to the Tx Group,
minus the IV
Since Control = Tx + IV, if our Tx Group has a
different mean than our Control Group, we can
attribute this to the IV
Hypothesis Testing w/Two
Samples
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Ex. If we want to know what effect fire has on
water, we have two groups, one that has water
over an open flame (the Tx Group), and one
that has water unheated (the Control Group). If
all other factors that could influence the heat of
water are kept constant between the two
groups (i.e. the water in the two groups has
the same salinity, the air pressure is the same,
etc.), if the water in the heated condition is
hotter, we can conclude that the difference is
due to the fire, and not the other factors that
we kept constant
Hypothesis Testing w/Two
Samples
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Assumptions of the Two-Samples TTest:
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1. Normally-distributed data

Like all other t-tests, and according to the
Central Limit Theorem, as long as sample sizes
are large, you can ignore this requirement
Hypothesis Testing w/Two
Samples
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2. Homogenous Variances (s2)
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As long as neither of your variances (s2) are
more than four times the other (you have two
variances, one for each sample) and your
samples have similar sample sizes (n), you’re
OK
If both: one of your variances is four or more
times the other and you have unequal sample
sizes, then you have to use another procedure
Hypothesis Testing w/Two
Samples
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3. Independent Samples

As stated previously, your groups have to be
independent – i.e. a subject can only be in one group
and the group cannot be yolked

Yolked Groups = you assign subjects to groups that look
similar on a variable or variables that you’re interested in
 Ex. You’re interested in sociability ratings, so since
subject 1 in Group A has a sociability rating of 45, the
next subject that has a similar rating you assign to
Group B
 Here, membership in Group B is dependent on the
subjects in Group A, i.e. the groups are not
independent
Hypothesis Testing w/Two
Samples
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How to compute a Two-Samples T-Test?
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Instead of subtracting μ from , you
subtract
from
X
X2
+t =
>X 1
-t = X 1 < X 2
X1
X2
Hypothesis Testing w/Two
Samples
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2
Also, instead of using s2, we use s p
2
2




n

1
s

n

1
s
1
2
2
s2  1
p
n1  n2  2

n for grp 1
variance for grp 1
n for grp. 2
variance for grp 2

This is what is called our Pooled Variance
Hypothesis Testing w/Two
Samples
2
2




n

1
s

n

1
s
2
1
2
2
s  1
n1  n2  2
p
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This formula averages the variances
from our two samples, however…

First it multiplies the variance by the
sample size [s2 x (n-1)], which gives more
importance to variances from larger
samples
Hypothesis Testing w/Two
Samples
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This is an example of what is called a
weighted average

Weighted Average = average where our
values to be average are multiplied by a
factor that we think is important (in this
case, n is this factor)
Hypothesis Testing w/Two
Samples

Therefore, our formula for our TwoSamples T-Test is:
t
X1  X 2
1 1 
s   
 n1 n2 
2
p
Hypothesis Testing w/Two
Samples
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Since we’re using two samples, our
df = n1  n2  2
Also, the form of our hypothesis changes:
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For a One-Tailed Test:
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H0 = (μ1 > μ2) (or visa-versa)
H1 = (μ1 ≤ μ2) (or visa-versa)
For a Two-Tailed Test:
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H0 = (μ1 = μ2)
H1 = (μ1 ≠ μ2)
Hypothesis Testing w/Two
Samples
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We would apply the Two-Samples T-Test
in the same way as previous tests:
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1. Identify H0 and H1
2. Calculate df and identify the critical t
3. Determine whether to use one- or twotailed test, determine what value of α to
use (usually .05), and identify the rejection
region(s) that the critical t is the boundary
of
Hypothesis Testing w/Two
Samples
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4. Calculate the variances for both samples
(s1 and s2), and use them to calculate the
pooled variance
5. Calculate the mean of both samples
6. Utilizing this information, calculate t
7. Compare your value of t to your critical
value and rejection region to determine
whether or not to reject H0
Hypothesis Testing w/Two
Samples
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Confidence Intervals:

The formula is the same as the one-sample
t-test, once again all that is different is that
we use two sample means instead of a
sample mean and a population mean, and
a pooled variance instead of a regular,
sample variance



CI = X 1  X 2 ± Critical t (two-tailed at p=.05)
x pooled standard deviation
Hypothesis Testing w/Two
Samples
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Example:

Much has been made of the concept of experimenter bias,
which refers to the fact that for even the most conscientious
experimenters there seems to be a tendency for the data to
come out in the desired direction. Suppose we use students
as experimenters. All the experimenters are told that
subjects will be given caffeine before the experiment, but
half the experimenters are told that we expect caffeine to
lead to good performance, and half are told that we expect
it to lead to poor performance. The dependent variable is
the number of simple arithmetic problems the subject can
solve in 2 minutes. The obtained data are as follows:
Hypothesis Testing w/Two
Samples
Expect Good
Performance
19
15
22
13
18
15
20
25
Expect Poor
Performance
14
18
17
12
21
21
24
14
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What are the Ho and H1?
What is the df and critical t?
What are your alpha, type of test (one- vs. two-tailed), and
rejection region(s)?
What is your t?
Will you reject or fail to reject the null hypothesis?
22
Hypothesis Testing w/Two
Samples
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Assuming a one-tailed test, H0 = (μ1 >
μ2); H1 = (μ1 ≤ μ2), where μ1 = “Expect
Good Performance”
df = 15; critical t(15) = 1.753
t = .587
Fail to reject the null hypothesis
Our data do not support the theory that
experimenter bias influences data.