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Independent-Samples T-Test: In-class example
General issue: Is there anything a person with Alzheimer’s disease can do to improve
their memory for their own personal history?
Specific research question: Will persons with AD who have been exposed on a regular
basis to cues for their personal history rate their own memories as better than those of
persons with AD who have not been exposed to these memory cues.
Experiment


Construct PowerPoint presentation that contain family photographs and audio
recollections.
After six weeks, give participants a measure of their perceived memory function.
Experimental Group
(Memory Cues)
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Control Group
(No memory cues)
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N1 = 10
M1 = 16.2
S1 = 2.49
N2 = 10
M2 = 9.9
S2 = 2.33
M1 – M2 = 16.2 – 9.9 = 6.3
Is this a difference that occurred by chance or is this a difference that is there because the
IV did something to make the people in group 1 different from the people in group 2?
HO: Persons with AD in the memory cues group do not have perceived memory scores
which are significantly greater than the perceived memory scores of persons with AD in
the control group. (The means for the two groups are different from each other by
chance)
H1: Persons with AD in the memory cues group have perceived memory scores which
are significantly greater than the perceived memory scores of persons with AD in the
control group. (The means are different from each other due to the effects of the
independent variable)
REMEMBER: The alternative hypothesis is the prediction of the researcher. The null
hypothesis is the opposite of the prediction.
α = .05
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This is a situation where there is no way of knowing for sure whether the null
hypothesis is true or whether the alternative hypothesis is true. We will never be
able to “prove” that either of them is true or false.
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
However, we can know exactly what the odds are that the null hypothesis is true.
We can know exactly what the odds are that the means for the two groups are
only different from each other by chance.

When we set our alpha level at .05, we’re saying that we’re only willing to reject
the null hypothesis if we can show that there is less than a five percent chance that
it’s true.
How can we know if the odds are less than 5% that HO is true?

We can do this using a test called an independent-samples t-test.

The number we’re making our decision about is the difference between the two
samples means. We can use SPSS to convert this number from a number in raw
score units into a number in standard score units. The symbol for this number is a
value for t.

Just like with a value for chi-square, if we get a value for t of zero, it tells us that
the number we get from our experiment (the difference between the means) is
exactly equal to what we’d expect to get if the null hypothesis were true. If we get
a difference between the means that’s a little bit different from zero, it’s pretty
likely that it’s just different from zero by chance. The further t is from zero (in
either direction), the less likely it is that the two sample means are that different
from each other just by accident.

At some point the value for t is far enough away from zero so that we get to the
point where the odds that it’s different from zero just by change become less than
5%. That’s the point where we ought to reject the null hypothesis. The critical
values for t table tells us how far from zero a value for t has to be in order to reject
the null hypothesis.

To know how to write a decision rule we need to think about the question we’re
trying to answer. In this example the researcher predicted that the mean for group
1 will be greater than the mean for group 2. In other words, they’re predicting that
the difference between the means will be in a particular direction. This is known
as a one-tailed test. It only makes sense to reject the null hypothesis if the mean
for group 1 really is greater than the mean for group 2. This means than we’re
predicting that the difference between the means will be a positive number. In
turn, this means that we’re predicting that the value for t will be a positive
number.

Decision rule: If t ≥ +1.73, reject HO.

Now we just need to use SPSS to find the observed value for t. According to
SPSS, the observed value for t is +5.85.
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5.85 is greater than 1.73, so our decision is to reject the null hypothesis.
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Because we’ve rejected the null hypothesis our conclusion is what we’ve already
written as the null hypothesis. That makes our conclusion:
Persons with AD in the memory cues group have perceived memory scores which are
significantly greater than the perceived memory scores of persons with AD in the control
group, t (18) = 5.85, p < .05.

Now, let’s say that we phrased our question differently. Suppose that the
researcher was only willing to bet that mean for the group getting the memory
cues would be different from the mean for the control group. Now the
researcher’s prediction is that the mean for group 1 could be either greater than or
less than the mean for group 2. This version of the test is called a two-tailed test.

The alternative hypothesis is the prediction of the researcher, so it is now: Persons
with AD in the memory cues group have perceived memory scores which are
significantly different than the perceived memory scores of persons with AD in
the control group.

The null hypothesis is now: Persons with AD in the memory cues group do not
have perceived memory scores which are significantly different than the
perceived memory scores of persons with AD in the control group.

If these are the null and alternative hypotheses, then the value for t could be
significant if it is far enough above or below zero. This means that there are now
two critical values for t and the decision rule would look like this:
If t ≤ -2.10 of if t ≥ +2.10, reject HO.
The observed value for t is the same (+5.85), so our decision is still to reject the null
hypothesis. Our decision is to accept our new alternative hypothesis and conclude that:
Persons with AD in the memory cues group have perceived memory scores which are
significantly different than the perceived memory scores of persons with AD in the
control group, t (18) = 5.85, p < .05.
Deciding between a one-tailed and a two-tailed test
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If the prediction of the researcher is that one sample mean will be different from
the other one, the test is two-tailed.
If the prediction is that one mean will be greater than the other mean, the test is
one tailed.
If the prediction is that one mean will be less than the other one, the test is onetailed.
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Where the observed and critical values for t come from
Above, we used a t-test to show us whether the odds were less than 5% or not that the
null hypothesis was true. But how can a t-test help us in knowing whether these odds are
less than 5% or not?

Statisticians can help us in this situation because they can show us what
differences between means are likely to look like if we were to keep doing the
same experiment over and over when the null hypothesis is true (i.e., when the
memory cues don’t help with memory).
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Pretend that you do the experiment and that you can know for sure that the
independent variable doesn’t have an effect. Working with the memory cues
doesn’t change people at all in terms of how good they perceive their memory to
be. If this is the case what would you guess the means for the two groups should
look like in relation to each other. You expect them to be the same number. That’s
because the experimenter didn’t do anything to make one group any different
from another group in terms of their memory function.
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But do the means for the two groups have to be equal to each other? No, the mean
of a sample is just an estimate – it’s not the real thing. So even in a situation
where two means should be the same number, you can’t really expect them to be.
They’ll almost certainly be at least a little bit different from each other just by
chance.

So how far do the means have to be from each other for us to bet that they’re not
that different by accident? That’s where the statisticians can help us. Imagine you
do the experiment once when the null hypothesis is true. There’s no reason, other
than chance, for the two means to different from each other, but let’s say that we
get a mean of 11.0 for group 1 and a mean of 10.5 for group 2. That gives us a
difference between the means of +.5. We can put that number where it goes on the
scale of possible differences between the mean. That’s one difference between the
means that was collected when the null hypothesis is true.
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Now pretend that you do the same experiment all over again. The null hypothesis
is still true. This time the mean of group 1 is 8.5 and the mean of group 2 is 9.0.
That makes the difference between the two means equal to -.5. Here’s where it
goes.

Now pretend that we did the same experiment over and over. Hundreds of times.
Thousands of times! We can now see how these differences between means pile
up around zero. Half are bigger than zero, just by accident. Half are less than zero,
just by accident. But the average of all of these number is exactly equal to the
number we’d expect it to be if the memory cues don’t work – zero.
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Across thousands of different experiments, here’s what differences between
means look like if the null hypothesis is true. Here’s the difference between
means we got from our experiment/ Are we willing to believe that our difference
between means belongs with these other ones or not? Is it believable that our
difference between means belongs with these other ones. If not, we’re deciding
that our difference between means must have been collected when the null
hypothesis is false.
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The name for this collection of differences between sample means is the
sampling distribution of the difference between means. It’s a collection of a
very large number of differences between means obtained when the null
hypothesis is true. We get one difference between means from each hypothetical
experiment we could do when the null hypothesis is true.
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The way the test works, every number in this hypothetical collection is converted
to a standard score. In other words, whenever we get a difference between means
we convert it to a standard score. You know what has to happen to convert any
number into a standard score. Take that number and divide it by the average of all
the number. In this case the average of all the numbers is always going to be zero.
This tells us how far one difference between the means is from zero. The value is
known as the standard error of the mean. From SPSS we learn that the standard
error of the mean for this data set is 1.08.
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Now divide this by the average amount that differences between means deviate
from zero. Now we’ve got our standard score. And the symbol for this standard
score is a value for t. (6.3 – 0)/1.08 = 5.85
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The critical value represents far away from a standard score of zero you have to
hit the start of the least likely 5% of numbers that make up the curve. With a onetailed test, we put all 5% of that bet on one side of the curve. With a two-tailed
test we split that 5% in half and put 2 ½% on one side and 2 ½% on the other side.
t=