T-Tests
PSY3200 Unit 5

Defining T-Tests

In this unit we will be taking the next step in our hypothesis testing. With a z score we compared a sample to a
known population where we knew the mean of that population and the standard deviation. A t-test is used when
you don't know that standard deviation ( Aron, Coups, & Aron, 2013) . Since the steps of hypothesis testing in this
unit are relatively unchanged we won't be going through them step by step but will rather be highlighting what is
different with a t-test. We have 3 different types of t-tests: single sample; dependent t-test, and independent t-test.

Single Sample T-Tests

A single sample t-test is used when you are comparing a sample to a population, you know the mean of the
population (µ), but unlike z scores, you do not know the variance of the population ( Aron et al., 2013) . Because
we don't know the variance, we will have to estimate it which we will be calling S2. In order to do this, we will need
what are called degrees of freedom. Degrees of freedom are the number of scores that are free to vary while still
following the same parameters ( Aron et al., 2013) . Degrees of freedom is calculated as n-1. So, if we have 20
participants in a study, the degrees of freedom for that study will be 19.

Let's explain this further of what exactly this means with the example of choosing a baseball line up. When a team
is batting there are 9 players that have to be chosen in a very specific order. When choosing that order there are
8 decisions you have to make. After you have chosen 8 positions, the 9th decision is made for you. So, there are
8 scores free to vary, while the final position is locked into place.

How degrees of freedom relate to estimating variance is in the final step of the variance calculation. Once we
subtract the mean from each value, square those answers, and then add them up (the sum of squares) we would
normally divide by n, but now we are going to divide by n-1 since we are estimating. The reason for this it allows
us to calculate an unbiased estimate ( Aron et al., 2013) . It is unbiased because by dividing by a lower number
we are just as likely to estimate a little too high as a little too low.

Once we have our estimated variance, we proceed very similar to z formula. You will take the estimated variance
of the population (S2) and divide it by the sample size which will give us S2 m, and then take the square root to

get Sm, the standard deviation of the distribution of means (or standard error). The formula for t
is very similar to that of Z:

The mean of the sample, minus the mean of the distribution of means divided by the standard deviation of the
means ( Aron et al., 2013) . Remember that standard deviation is created using an estimated variance for the
population.

Other than that, most steps of hypothesis testing are similar to z scores with the exception of the cutoff score. For
that we use a new table which was provided for you (). To figure out the t cutoff score, you need to narrow down if
it is a one or two tailed test, then select the alpha level, and the degrees of freedom (Nolan & Heinzen, 2017). As
a reminder two tailed tests have both a + and a - for cutoff scores, while one tailed tests are one or the other.

Dependent T-Tests

The second type of t-test is called a dependent t-test (often called a paired sample t-test). This type of test is used

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when you are comparing a single sample to themselves ( Aron et al., 2013) . Here you will also not know the
population variance. This type of test is commonly seen as a before and after test. The sample is measured, then
exposed to some IV, which then tested again to see if it played a significant effect. How these tests works are very
similar to single sample t-tests, with a couple minor differences. The first difference is every individual in the
sample has two scores: a before score and an after score ( Aron et al., 2013) . Your first step is to calculate the
difference scores by taking the after score and subtracting the before scores. Once you've done this you have a
single set of scores and the average of these difference scores is your M. Once you've done this, you would
proceed the same as you would for a single sample t-test: (1) estimate the population variance, (2) calculate the
variance of the distribution of means; and (3) the standard deviation of the distribution of means.

The other aspect of these tests that are different is the mean of the population is zero. The reason for this is mean
is the average of the comparison group; the one that did not receive the IV. If the IV did not take place then there
should not be a difference between the before and after score, thus the difference between them should be zero (
Aron et al., 2013) .

Independent T-Tests

The final type of t-test is an independent t-test, which is used when you are comparing two samples from two
different populations to each other ( Aron et al., 2013) . The most practical example would be an experimental
group compared to a control group. Since we are dealing with two different populations, we will have two different
distribution of means for the comparison distribution, but instead of using one, we will combine them into a single
"distribution of differences between means." To explain how this works, let's use the example of investigating the
difference between math grades of freshmen and seniors. I would take a sample of freshmen and get their
average grade; a sample of seniors and get their average age; and then get the difference between those
averages. I would do this again and again, and eventually I would have a set of scores that is made up of
differences between means of the different populations ( Aron et al., 2013) . Hence a "distribution of differences."

For these problems we will once again be calculating t, but there is a new formula and multiple new steps to
follow. The first thing you would have to calculate are the means and estimated variances for each group. Next
your goal is to calculate what is called the pooled estimate of the population variance. To get this you take the
estimated variance for each group and multiply it by the ratio of degrees of freedom for that group over the
degrees of freedom total (added together degrees of freedom) ( Aron et al., 2013) . As a hint, this answer will
always be a number in between the two estimated variances (S2). Once you have this value, you would divide
this answer be the sample size of each group to get the variance of the distribution of means for each group (S2

m). You would add those together to get the variance of the distribution of differences between means (S2

difference) and finally take the square root to get the standard deviation of the distribution of differences between
means (Sdifference). Now that we have our distribution of differences between means we can calculate t using the
formula:

M1 is the mean of population 1, M2 is the mean of population 2, and Sdifference is the standard deviation of the
distribution of differences between means ( Aron et al., 2013) . Obtaining the cutoff score is very similar to the
previous two types of t-tests, the only difference is we would use the degrees of freedom total. The remaining
steps are the same as all other types of hypothesis testing.

Using SPSS for T-Tests

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For this unit we will also be calculating t-tests using the SPSS program. This program is much more efficient for
this type of statistical test. While it is simpler to use and convenient since one only has to plug in the numbers and
tell the program the correct sequence, it is important to know the steps of doing the problems by hand, as
computer programs can change, but the math will always be constant.

For a single sample t-test, enter your data in a single column. You then click analyze -> compare means -> one
sample t-test. You then select the column of data you wish to analyze for "test variable" and for "test values" you
put the given µ. Under options you can select things such as measures of central tendency and variability. Finally
click paste to compute the results.

For a dependent t-test, enter the before and after data in two different columns. Click analyze -> compare means
-> paired samples t-test. You can then select the two columns of data and hit the arrow button to analyze it (it will
give you the option to switch which data is before and which is after if needed). You will then click paste to
compute the results ( Aron et al., 2013) .

For an independent t-test, you have all the data typed into a single column. In the next column place a "1" next to
any number coming from the first population, and a "2" next to any number coming from the second population.
Click analyze -> compare means -> independent samples t-test. Click the column with your scores for test
variable (use the arrow to bring it over) and click the column with the group numbers for the grouping variable
(use the arrow to bring it over). Once you've done this, click OK and your data will be computed ( Aron et al.,
2013) .

It is important to know that the program can do a great many more things such as naming data groups, controlling
decimal numbers, naming variables etc. There are some fantastic online tutorials on some of these minor details
that a simple google search will help you accomplish ( Aron et al., 2013) .

Interpreting SPSS Outputs

The final thing to discuss is how to interpret an SPSS output file. You are given several columns of data when a
set of data is computed. For a t-test, the three most important are: t; df; and sig. The t is the t statistic, what you
would get if you calculated t by hand. df is your degrees of freedom. sig is your significance value, and this one
is the most important. Using SPSS, this value will tell you if your result had a significant difference or not ( Aron et
al., 2013) . For example, if it was a single sample t-test, then your sample is significantly different from the
population. For a dependent t-test, it means there was a significant difference between the before and after score,
so the independent variable changed the peoples' scores. For an independent t-test, it means there was a
significant difference between the two populations you tested.

How do you determine if it was significant or not? If the number in the sig column (which we call a p value) is 0.05
or below, then yes it was significant, and we would reject the null hypothesis ( Aron et al., 2013) . If it was above
0.05 then no, it was not a significant difference and we would fail to reject the null hypothesis. When we are
reporting a t statistic that was significant, we use the following format: t(df)=[t value], p<0.05. So, if t was 5.45, and
there were 20 people in my study, it would read: t(19) = 5.45, p<0.05. The one thing to note is if the p value is
lower than 0.01, we would use that instead of 0.05.

(CSLO 1, CSLO 2, CSLO 3, CSLO 4, CSLO 5)

References

Aron, A. Coups, E.J. & Aron, E. (2013) Statistics for Psychology (6th ed.) Chapter 7-8.
Nolan, S. & Heinzen, T. (2017) Statistics for the Behavioral Sciences (4th ed.) Appendix B.

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