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The One-Sample t-Test
Advanced Research Methods in Psychology
- lecture -
Matthew Rockloff
1
A brief history of the t-test



Just past the turn of the nineteenth century, a
major development in science was fermenting
at Guinness Brewery.
William Gosset, a brewmaster, had invented a
new method for determining how large a
sample of persons should be used in the
taste-testing of beer.
The result of this finding revolutionized
science, and – presumably - beer.
2
A brief history of the t-test (cont.)

In 1908 Gosset published his
findings in the journal Biometrika
under the pseudonym ‘student.’
This is why the t-test is often called
the ‘student’s t.’
3
A brief history of the t-test (cont.)
Folklore: Two stories circulate for the reason why
Gosset failed to use his own name.
 1: Guinness may have wanted to keep their
use of the ‘t-test’ secret. By keeping Gosset
out of the limelight, they could also protect
their secret process from rival brewers.
 2: Gosset was embarrassed to have his name
associated with either: a) the liquor industry,
or b) mathematics.

But seriously, how could that be?
4
Thus ???
“Beer is the cause of –
and solution to –
all of life's problems.”
(Homer Simpson)
5
When to use the one-sample t-test




One of the most difficult aspects of
statistics is determining which procedure to
use in what situation.
Mostly this is a matter of practice.
There are many different rules of thumb
which may be of some help.
In practice, however, the more you
understand the meaning behind each of the
techniques, the more the choice will become
obvious.
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Example problem 2.1


Example illustrates the calculation of
the one-sample t-test.
This test is used to compare a list of
values to a set standard.



What is this standard?
The standard is any number we choose.
As illustrated next, the standard is
usually chosen for its theoretical or
practical importance.
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Example 2.1 (cont.)



Intelligence tests are constructed such that
the average score among adults is 100
points.
In this example, we take a small sample of
undergraduate students at Thorndike
University (N = 6), and try to determine if the
average of intelligence scores for all
students at the university is higher than 100.
In simple terms, are the university
students smarter than average?
8
Example 2.1 (cont.)

The scores obtained from the 6 students
were as follows:
X
Person 1:
Person 2:
Person 3:
Person 4:
Person 5:
Person 6:
110
118
110
122
110
150
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Example 2.1 (cont.)
Research Question
On average, do the population of
undergraduates at Thorndike
University have higher than average
intelligence scores (IQ  100)?
10
Example 2.1 (cont.)

First, we must compute the mean (or
average) of this sample:
=

   110  118  110  122  110  150  120
n
6
In the above example, there is some new
mathematical notation. (See next slide)
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Example 2.1 (cont.)
=

   110  118  110  122  110  150  120
n
6
First, a symbol that denotes the mean
of all Xs or intelligence scores.
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Example 2.1 (cont.)
=

   110  118  110  122  110  150  120
n
6
The second part of the equation shows
how this quantity is computed.
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Example 2.1 (cont.)
=

   110  118  110  122  110  150  120
n
6
The sigma symbol (  ) tells us to sum
all the individual Xs.
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Example 2.1 (cont.)
=

   110  118  110  122  110  150  120
n
6
Lastly, we must divide by ‘n’,
that is: the number of observations.
15
Example 2.1 (cont.)
=

   110  118  110  122  110  150  120
n
6
Notice, these 6 people have higher than
average intelligence scores (IQ  100).
16
Example 2.1 (cont.)




However, is this finding likely
to hold true in repeated samples?
What if we drew 6 different people from
Thorndike University?
A one-sample t-test will help answer this
question.
It will tell us if our findings are ‘significant’,
or in other words, likely to be repeated if we
took another sample.
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Example 2.1 (cont.)
Computing the sample variance
 
110
118
110
122
110
150
120
120
120
120
120
120
   (   )
-10
-2
-10
2
-10
30
2
100
4
100
4
100
900
s
2
x
 (   )

n
2
 201. 3
18
Example 2.1 (cont.)
Computing the sample variance

(



)




2
s 
110 x
118
110
122
110
150
120
120
120
120
120
120
n
2
   (   )
 201. 3 100
-10
-2
-10
2
-10
30
2
4
100
4
100
900
s
2
x
 (   )

n
2
 201. 3
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Example 2.1 (cont.)
Computing the sample variance




To get the third column we take each
individual ‘X’ and subtract it from the
mean (120).
We square each result to get the fourth
column.
Next, we simply add up the entire
fourth column and divide by our
original sample size (n = 6).
The resulting figure, 201.3, is the
sample variance.
20
Example 2.1 (cont.)
Computing the sample variance
Important:
All sample variances
are computed this way!
We always take the mean;
subtract each score from the mean;
square the result;
sum the squares;
and divide by the sample size
(how many numbers, or rows we have).
21
Example 2.1 (cont.)
Computing the sample variance


Now that we have the mean (X = 120) and
2
the variance ( s x  201. 3 ) of our sample, we
have everything needed to compute whether
the sample mean is ‘significantly’ above the
average intelligence.
In the formula that follows, we use a new
symbol mu (  ) to indicate the population
standard value (  = 100 ) against which we
compare our obtained score (X = 120).
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Example 2.1 (cont.)
Computing the sample variance

120  100
t

 3.152
2
201.333
sx
6

1
n 1
Our sample has ‘n = 6’ people, so the degrees of
freedom for this t-test are:
dn = n – 1 = 5
This degrees of freedom figure will be used later
in our test of significance.
23
And now for something
completely different …
Let’s take a break from computations, and
talk about ‘the big picture’
Now comes the conceptually tricky part.
 Remember that a normal bell-curve
distribution is a chart that shows
frequencies (or counts).
 If we measured the weight of four male
adults, for example, we might find the
following:
 Person 1 = 70 kg, Person 2 = 75 kg, Person 3
= 70 kg, Person 4 = 65 kg

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The ‘big picture’


Person 1 = 70 kg, Person 2 = 75 kg, Person
3 = 70 kg, Person 4 = 65 kg
Plotting a count of these ‘weight’ data,
we find a normal distribution:
Count 2
1
X
X
X
X
65
70
75 kg
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The ‘big picture’ (cont.)

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As it turns out, the ‘t’ statistic has its own distribution,
just like any other variable.
Let’s assume, for the moment, that the mean IQ of the
population in our example is exactly 100.
If we repeatedly sampled 6 people and calculated
a ‘t’ statistic each time, what would we find?
If we did this 4 times, for example, we might find:
Count 2
1
X
X
X
X
-1
0
1
t - statistic
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The ‘t’ statistic (cont.)


Most often our computed ‘t’
should be around 0. Why?
Because the numerator, or top part of
the formula for t is:    .
If our first sample of 6 people is truly
representative of the population, then
our sample mean should also be 100,
and therefore our computed t should
be  (see next slide)
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The ‘t’ statistic (cont.)
t
100  100
s
2
x
0
n 1
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The ‘t’ statistic (cont.)

Of course, our repeated samples
of 6 people will not always have
exactly the same mean
as the population.

Sometimes it will be a little higher, and
sometimes a little lower.

The frequency with which we find a t larger
than 0 (or smaller than 0) is exactly what the
t-distribution is meant to represent
 (see next slide)
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The ‘t’ statistic (cont.)
30
Read this part over and over,
and think about it. This is the tricky bit.



In our GPA example, the actual ‘t’ that
we calculated was 3.152, which is
certainly higher than 0.
Therefore, our sample does not look
like it came from a population with a
mean of 100.
Again, if our sample did come from
this population, we would most often
expect a computed ‘t’ of 0
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The ‘t’ statistic (cont.)


How do we know when
our computed ‘t’ is
very large in magnitude?
Fortunately, we can calculate how
often a computed sample ‘t’ will be far
from the population mean of t = 0
based on knowledge of the
distribution.
32
The ‘t’ statistic (cont.)



The critical values of the t-distribution show
exactly how often we should find computed
‘t’s of large magnitude.
In a slight wrinkle, we need the degrees of
freedom (df = 5) to help us make this
determination.
Why? If we sample only a few people our
computed ‘t’s are more likely to be very
large, only because they are less
representative of the whole population.
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And now, back to the computation…




We need to find the ‘critical value’ of
our t-test.
Looking in the back of any statistics
textbook, you can find a table for critical
values of the t-distribution.
Next, we need to determine whether we
are conducting a 1-tailed or 2-tailed t-test.
Let’s refer back to the research question:
34
Example 2.1 (again)
Research Question
On average, do the population of
undergraduates at Thorndike
University have higher than average
intelligence scores (IQ  100)?
35
Example 2.1 (cont.)




This is a 1-tailed test, because we are asking
if the population mean is ‘greater’ than 100.
If we had only asked whether the
intelligence of students were ‘different’ from
average (either higher or lower) then the test
would be 2-tailed.
In the appendixes of your textbook, look at
the table titled, ‘critical values of the tdistribution’.
Under a 1-tailed test with an Alfa-level of
and degrees of freedom df = 5, and you
should find a critical value (C.V.) of t = 2.02.
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Example 2.1 (cont.)
Is our computed t = 3.152
greater than the C.V. = 2.02?
Yes!
Thus we reject the null hypothesis
and live happily ever after.
Right?
Not so fast.
What does this really mean?
37
Example 2.1 (still)



We assume the null hypothesis when
making this test.
We assume that the population mean
is 100, and therefore we will most often
compute a t = 0.
Sometimes the computed ‘t’ might be a
bit higher and sometimes a bit lower.
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What does the ‘critical value’ tell us?

Based on knowledge of the distribution
table we know that 95% of the time, in
repeated samples, the computed ‘t’
statistics should be less than 2.02.

That’s what the critical value tells us.

It says that when we are sampling 6 persons
from a population with mean intelligence
scores of 100, we should rarely compute a ‘t’
higher than 2.02.
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What happens if
we do calculate a ‘t’ greater than 2.02?

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

Well, we can be pretty confident that our
sample does not come from a population
with a mean of 100!
In fact, we can conclude that the population
mean intelligence must be higher than 100.
How often will we be wrong in this
conclusion?
If we do these t-tests a lot, we’ll be wrong
5% of the time. That’s the Alfa level (or 5%).
40
Conclusions in APA Style
How would we state our
conclusions in APA style?
The mean intelligence score of
undergraduates at Thorndike University
(M = 120) was significantly higher than
the standard intelligence score (M =
100), t(5) = 3.15, p < .05 (one-tailed).
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Statistical inference

You should notice that the conclusion
makes an inference about the population of
students from Thorndike University based
on a small sample.

This is why we call this type of a test
‘statistical inference.’
We are inferring something about the
population based on only a sample of
members.

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Example 2.1 Using SPSS

First, variables must be setup in the variable
view of the SPSS Data Editor as detailed in
the previous chapter:
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Example 2.1 Using SPSS (cont.)

Next, the data must be entered in the data view of
the SPSS Data Editor:
44
Instructions
for the Student Version of SPSS



If you have the student version of
SPSS, you must run all procedures
from the pull-down menus.
Fortunately, this is easy for the onesample t-test.
First select the correct procedure from
the ‘analyze’ menu  see next slide 
45
The ‘analyze’ menu
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The ‘test variable’


Next, you must move the ‘test variable’, in this
case IQ, into the right-hand pane by pressing
the arrow button and change the ‘test-value’
to 100 (our standard for comparison).
Lastly, click ‘OK’ to view the results:
47
Instructions for
Full Version of SPSS (Syntax Method)




An alternate method for obtaining the same
results is available to users of the fullversion of SPSS.
This method, known as ‘syntax’, is
described here, because many common and
useful procedures in statistics are only
available using the syntax method.
Users of the student version may wish to
skip ahead to ‘Results from the SPSS
Viewer.’
To use syntax, first you must open the
syntax window from the ‘file’ menu: 
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The ‘file’ menu
49
SPSS syntax

The following is generic syntax
for the one-sample t-test:
 t-test testval=TestValue
 /variables=TestVariable.

The SPSS syntax above requires that you
substitute two values.
First, you need the ‘TestValue’ against which
you are judging your sample.
In example 2.1, this standard is ‘100.’


50
TestVariable

Next, you must substitute the ‘TestVariable,’
as shown below:
51
Results from SPSS Viewer

After selecting ‘Run – All’ from the menu, the
results will appear in the SPSS output window:
52
SPSS calculations




When using SPSS, we no longer have a
critical value to compare our calculated
t-value.
Instead, SPSS calculates an exact
probability value associated with the ‘t.’
As a consequence, when writing the results
we simply substitute this exact value, rather
than using the less precise ‘p < .05’ (per our
hand-calculations above).
Notice that SPSS calls p-values ‘Sig.,’ which
stands for significance.
53
SPSS calculations

NOTE: SPSS only gives us the
p-value for a 2-tailed t-test.

In order to convert this value into a onetailed test, per our example, we need to
divide this ‘sig (2-tailed)’ value in half
(e.g., .033/2=.02, rounded).
Why?
In short, one-tailed t-tests are twice as
powerful, because we simply assume that
the results cannot be different in the
direction opposite to our expectations.


54
Conclusions in APA Style
Focusing attention on the bold
portion of the output, we can
re-write our conclusion in APA style:
The mean intelligence score of
undergraduates at Thorndike University
(M = 120) was significantly higher than
the standard intelligence score (M =
100), t(5) = 3.15, p = .02 (one-tailed).
55
Big t  Little p ?



Remember from a previous session that
every t-value that we might calculate is
associated with a unique p-value.
In general, t-values which are large in
absolute magnitude are desirable, because
they help us to demonstrate differences
between our computed mean value and the
standard.
Values of t that are large in absolute
magnitude are always associated with small
p-values.
56
Significance

According to tradition in psychology,
p-values which are lower than .05 are
significant, meaning that we will likely
still find differences if we collected
another sample of participants.

When using SPSS we are no longer
confronted with a ‘critical value.’
Instead, we can simply observe that
the p-value is less than ‘.05.’
57
Accepting the null hypthosis

The conclusion as to whether to reject
the null hypothesis will be the same in
each circumstance; whether computed
by-hand
or by-computer.
58
The One-Sample t-Test
Advanced Research Methods in Psychology
Week 1 lecture
Matthew Rockloff
59