PSY 307 – Statistics for the
Behavioral Sciences
Chapter 13 – Single Sample t-Test
Chapter 15 -- Dependent Sample tTest
Midterm 2 Results
Score
Grade
N
45-62
A
7
40-44
B
3
34-39
C
4
29-33
D
4
0-28
F
3
The top score on the exam
and for the curve was 50 – 2
people had it.
Student’s t-Test

William Sealy Gossett published
under the name “Student” but was
a chemist and executive at Guiness
Brewery until 1935.
What is the t Distribution?



The t distribution is the shape of the
sampling distribution when n < 30.
The shape changes slightly
depending on the number of
subjects in the sample.
The degrees of freedom (df) tell you
which t distribution should be used
to test your hypothesis:

df = n - 1
Comparison to Normal Distribution




Both are symmetrical, unimodal,
and bell-shaped.
When df are infinite, the t
distribution is the normal
distribution.
When df are greater than 30, the t
distribution closely approximates it.
When df are less than 30, higher
frequencies occur in the tails for t.
The Shape Varies with the df (k)
Smaller df produce larger tails
Comparison of t Distribution and
Normal Distribution for df=4
Finding Critical Values of t




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Use the t-table NOT the z-table.
Calculate the degrees of freedom.
Select the significance level (e.g.,
.05, .01).
Look in the column corresponding to
the df and the significance level.
If t is greater than the critical value,
then the result is significant (reject
the null hypothesis).
Link to t-Tables
http://www.statsoft.com/textboo
k/sttable.html
Calculating t


The formula for t is the same as
that for z except the standard
deviation is estimated – not known.
Sample standard deviation (s) is
calculated using (n – 1) in the
denominator, not n.
Confidence Intervals for t

Use the same formula as for z but:




Substitute the t value (from the ttable) in place of z.
Substitute the estimated standard error
of the mean in place of the calculated
standard error of the mean.
Mean ± (tconf)(sx)
Get tconf from the t-table by
selecting the df and confidence level
Assumptions



Use t whenever the standard
deviation is unknown.
The t test assumes the underlying
population is normal.
The t test will produce valid results
with non-normal underlying
populations when sample size > 10.
Deciding between t and z

Use z when the population is normal
and s is known (e.g., given in the
problem).


Use t when the population is normal
but s is unknown (use s in place of s).
If the population is not normal,
consider the sample size.


Use either t or z if n > 30 (see above).
If n < 30, not enough is known.
What are Degrees of Freedom?



Degrees of freedom (df) are the
number of values free to vary given
some mathematical restriction.
Example – if a set of numbers must
add up to a specific toal, df are the
number of values that can vary and
still produce that total.
In calculating s (std dev), one df is
used up calculating the mean.
Example

What number must X be to make
the total 20?
5
10
7
X
20
100
200
300
X
20
Free to vary
Limited by the constraint
that the sum of all the
numbers must be 20
So there are 3 degrees of freedom in this example.
A More Accurate Estimate of s



When calculating s for inferential
statistics (but not descriptive), an
adjustment is made.
One degree of freedom is used up
calculating the mean in the
numerator.
One degree of freedom must also
be subtracted in the denominator to
accurately describe variability.
Within Subjects Designs

Two t-tests, depending on design:



t-test for independent groups is for
Between Subjects designs.
t-test for paired samples is for Within
Subjects designs.
Dependent samples are also called:



Paired samples
Repeated measures
Matched samples
Examples of Paired Samples

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Within subject designs
Pre-test/post-test
Matched-pairs
Independent samples – separate
groups
Dependent Samples



Each observation in one sample is
paired one-to-one with a single
observation in the other sample.
Difference score (D) – the
difference between each pair of
scores in the two paired samples.
Hypotheses:


H0: mD = 0
H1: mD ≠ 0
mD ≤ 0
mD > 0
Repeated Measures


A special kind of matching where
the same subject is measured more
than once.
This kind of matching reduces
variability due to individual
differences.
Calculating t for Matched Samples


Except that D is used in place of X,
the formula for calculating the t
statistic is the same.
The standard error of the sampling
distribution of D is used in the
formula for t.
Degrees of Freedom



Subtracting values for two groups
gives a single difference score.
The differences, not the original
values, are used in the t calculation,
so degrees of freedom = n-1.
Because observations are paired,
the number of subjects in each
group is the same.
Confidence Interval for mD



Substitute mean of D for mean of X.
Use the tconf value that corresponds
to the degrees of freedom (n-1) and
the desired a level (e.g., 95%= .05
two tailed).
Use the standard deviation for the
difference scores, sD.

Mean D ± (tconf)(sD)
When to Match Samples



Matching reduces degrees of
freedom – the df are for the pair,
not for individual subjects.
Matching may reduce generality of
the conclusion by restricting results
to the matching criterion.
Matching is appropriate only when
an uncontrolled variable has a big
impact on results.
Deciding Which t-Test to Use

How many samples are there?


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Just one group -- treat as a population.
One sample plus a population is not
two samples.
If there are two samples, are the
observations paired?


Do the same subjects appear in both
conditions (same people tested twice)?
Are pairs of subjects matched (twins)?
Population Correlation Coefficient



Two correlated variables are similar
to a matched sample because in
both cases, observations are paired.
A population correlation coefficient
(r) would represent the mean of r’s
for all possible pairs of samples.
Hypotheses:


H0: r = 0
H1: r ≠ 0
t-Test for Rho (r)


Similar to a t–test for a single
group.
Tests whether the value of r is
significantly different than what
might occur by chance.

Do the two variables vary together by
accident or due to an underlying
relationship?
Formula for t
t
r  r hyp
1 r
n2
2
Standard error of
prediction
Calculating t for Correlated Variables


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Except that r is used in place of X,
the formula for calculating the t
statistic is the same.
The standard error of prediction is
used in the denominator to
calculate the standard deviation.
Compare against the critical value
for t with df = n – 2 (n = pairs).
Importance of Sample Size

Lower values of r become significant
with greater sample sizes:

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As n increases, the critical value of t
decreases, so it is easier to obtain a
significant result.
Cohen’s rule of thumb
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.10 = weak relationship
.30 = moderate relationship
.50 = strong relationship