t-Test
Comparing Means From Two Sets
of Data
Steps For Comparing Groups
Assumptions of t-Test
Dependent variables are interval or ratio.
The population from which samples are
drawn is normally distributed.
Samples are randomly selected.
The groups have equal variance
(Homogeneity of variance).
The t-statistic is robust (it is reasonably
reliable even if assumptions are not fully
met.
Computing Confidence Intervals
We can determine the probability that a population mean
lies between certain limits using a sample mean.
With inferential statistics we reverse this process and
determine the probability that a random sample drawn
from a specific population would differ by an observed
result.
t Values
Critical value
decreases if N is
increased.
Critical value
decreases if alpha
is increased.
Differences
between the means
will not have to be
as large to find sig
if N is large or
alpha is increased.
Probability that a sample came from a population?
Using the standard error we compute the probability that
two means come from the same population.
If Z or t exceed the level of significance we conclude that
the sample was


Not drawn from the population or
Has been modified so that it no longer represents the population
Relationship between t Statistic and Power
To increase power:




Increase the difference
between the means.
Reduce the variance
Increase N
Increase α from α = .01 to
α = .05
Does Volleyball Serve Training Improve Serving Ability?
Population mean = 31, sd =
7.5.
30 students given serve
training. Following training
mean = 35, sd = 8.3.
Critical Z = 1.96
Probability is greater than
99 to 1 that the mean did
not come from original
population.
The training was effective.
Volleyball Example Using t-statistic
Critical value of t(29)= 2.045, p = 0.05
Since obtained t > critical value these
means are statistical different.
Comparing Two Independent Samples
Independent samples (males, females),
(swimmers, runners).
Must be different subjects in each group.
Independent t Test
If the t statistic is greater than the critical
value we
Conclude the independent variable had a
significant effect
And we reject chance as the cause of the
mean difference.
Effects of Verbal Lesson of Basketball Shooting Skill
Critical value of t(120) = 1.98, p = 0.05
Since our obtained t(98) = -1.36 is NOT greater than the critical value we
ACCEPT the Null Hypothesis. The training had no effect upon shooting skill.
Note: The sign +/- of t does not matter.
Does Positive
Reinforcement
Affect Bowling?
Critical value t(40) = 2.201, p = 0.05
Since obtained t > critical t
We reject the Null and state that
positive reinforcement significantly
improves bowling ability.
Summary Table for Effects of Praise on Bowling
The t-test With Unequal N
When you have unequal numbers of subjects in
each group the statistic uses a different equation to
estimate the standard error of the differences
between groups.
The t-test With Unequal N
Critical value of t(16) = 2.120, p = .05. The groups are significantly different.
Dependent or Paired t-test
Note that the equation uses the correlation between pre and post samples.
The Dependent t-test is more powerful that the
Independent Groups t-test.
Dependent or Paired t-test
The same subjects are in
each group
(DEPENDENT or
PAIRED t-test).
Critical value t(29) =
2.045, p = 0.05
The groups ARE
SIGNIFICANTLY
Different.
Note: the correction
formula adjusts the
variance between groups.
Since the same subjects
are in each group you
can expect less variance.
Repeated Measures
experiments are more
powerful than
independent groups
Does a Bicycle Tour Affect Self-Esteem?
Are these differences MEANINGFUL????
Critical value of t(60) = 2.000, p = 0.05, so there is a
significant difference. BUT DOES IT MEAN ANYTHING???
The Magnitude of the Difference (Size of Effect)
Omega squared can be used to determine the
importance, or usefulness of the mean difference.
ω2 is the percentage of the variance (diff between
means) that can be explained by the independent
variable.
In this case the low-back and hip study explains
21% of variance between the means (pre & post).
Cohen’s Effect Size
Effect size of .2 is small, .5 moderate, .8 large
The control group is used to compute SD
because it is not contaminated by the treatment
effect.
The Percent Change is also useful in evaluating if a change
is meaningful.
Before doing an experiment you should know
what Percent Change would be considered
meaningful.
For an Olympic athlete, a 1% (meaningful)
improvement can be the difference between
winning and losing.
For an untrained individual a 1% improvement
would probably be meaningless.
Practical & Meaningful Significance
If two means are significantly different, that does
not imply that they are practical.
If two means are NOT statistically significant,
that does not imply that their differences are not
practical.
Use ω2, Effect Size and Percent Change to
evaluate the meaningfulness of an outcome.
Type I and Type II Errors
Type I Error: Stating that there is a difference
when there isn’t.
Type II Error: Stating there is no difference when
there is one.
We can never know if we have made a Type I or II error.
Statistics only provide the probability of making a Type I or
II error.
The critical factor in this decision is the consequence of
being wrong.
The confidence level should be set to protect against the
most costly error.
Which is worse: to accept the null hypothesis when it is
really false or to reject it when it is really true?
Two Tailed Test: Null No Difference.
One Tail Test: Null A > B. More Powerful, easier to find
differences.
Power: the ability to detect differences if they exist.
Statistical Power
1.
2.
3.
4.
Power ( 1 - β ) depends upon:
Alpha [Zα (.10) = 1.65, Zα (.05) = 1.96]
Difference between the means.
Standard deviations between the two
groups.
Sample size N.
To Increase Power
Increase alpha, Power for α = .10 is
greater than power for α = .05
Increase the difference between means.
Decrease the sd’s of the groups.
Increase N.
Calculation of Power
From Table A.1 Zβ of
.54 is 20.5%
Power is
20.5% + 50% = 70.5%
In this
example
Power (1 - β )
= 70.5%
Calculation of Sample
Size to Produce a
Given Power
Compute Sample Size N for a Power of .80 at p = 0.05
The area of Zβ must be 30% (50% + 30% = 80%) From Table A.1
Zβ = .84
If the Mean Difference is 5 and SD is 6 then 22.6 subjects would
be required to have a power of .80
Calculation of Sample Sized Need to Obtain a Desired
Level of Power
PSD
30
Alpha
Newtons
1.96
this is p=.05
80
0.84
these are beta values
90
1.28
95
1.645
Beta
Power
Stdev
80
90
95
30
16
21
26
20
7
9
12
10
2
2
3
These values in red are the N
needed based on your
PSD.
Power
Research performed with insufficient
power may result in a Type II error,
Or waste time and money on a study that
has little chance of rejecting the null.
In power calculation, the values for mean
and sd are usually not known beforehand.
Either do a PILOT study or use prior
research on similar subjects to estimate
the mean and sd.
Independent t-Test
For an Independent
t-Test you need a
grouping variable to
define the groups.
In this case the
variable Group is
defined as
1 = Active
2 = Passive
Use value labels in
SPSS
Independent t-Test: Defining
Variables
Be sure to
enter value
labels.
Grouping variable GROUP, the level of
measurement is Nominal.
Independent t-Test
Independent t-Test: Independent &
Dependent Variables
Independent t-Test: Define Groups
Independent t-Test: Options
Group Statistics
Ab_Error
Group
Active
Pas sive
N
10
10
Mean
2.2820
1.9660
Std. Deviation
1.24438
1.50606
Std. Error
Mean
.39351
.47626
Independent t-Test: Output
Independent Samples Test
Levene's Tes t for
Equality of Variances
F
Ab_Error
Equal variances
ass umed
Equal variances
not as sumed
.513
Sig.
.483
t-tes t for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
.511
18
.615
.31600
.61780
-.98194
1.61394
.511
17.382
.615
.31600
.61780
-.98526
1.61726
Assumptions: Groups have equal variance [F =
.513, p =.483, YOU DO NOT WANT THIS TO
BE SIGNIFICANT. The groups have equal
variance, you have not violated an assumption
of t-statistic.
Are the groups
different?
t(18) = .511, p = .615
NO DIFFERENCE
2.28 is not different
from 1.96
Dependent or Paired t-Test: Define Variables
Dependent or Paired t-Test: Select PairedSamples
Dependent or Paired t-Test: Select Variables
Dependent or Paired t-Test: Options
Paired Samples Statistics
Pair
1
Pre
Pos t
Mean
4.7000
6.2000
N
10
10
Std. Error
Mean
.66750
.90431
Std. Deviation
2.11082
2.85968
Dependent or Paired tTest: Output
Paired Samples Correlations
N
Pair 1
Pre & Pos t
10
Correlation
.968
Sig.
.000
Paired Samples Test
Paired Differences
Pair 1
Pre - Post
Mean
-1.50000
Std. Deviation
.97183
Std. Error
Mean
.30732
95% Confidence
Interval of the
Difference
Lower
Upper
-2.19520
-.80480
t
-4.881
Is there a difference between pre & post?
t(9) = -4.881, p = .001
Yes, 4.7 is significantly different from 6.2
df
9
Sig. (2-tailed)
.001