Transcript t Test for a Single Sample

Chapter 8
Introduction to the t
Test
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
t Test for a Single Sample
• Used to compare the mean of a sample with
a population for which the mean is known
but the variance is unknown
• Unlike previous methods, one must now
estimate the population variance from the
scores in the sample
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
t Test for a Single Sample
• Estimating population
variance from sample
scores
– Variance in sample
generally slightly smaller
than population
• Sample is a biased
estimate of population
• So, divide by N-1 rather
than N to correct for bias
– N-1 is known as the
“degrees of freedom,” the
number of scores that are
free to vary
S
2
(X  M )


2
N 1
S 
2
SS

N 1
2
(
X

M
)

df
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
SS

df
t Test for a Single Sample
• Because variance is estimated,
comparison distribution is not a normal
curve
– t distribution
– Like a normal curve
• Bell-shaped
• Unimodal
• Symmetrical
– But has more scores at the extremes
(i.e., heavier tails) and varies somewhat
according to degrees of freedom
• Sample mean thus has to be slightly
more extreme to be significant with a t
distribution than with a normal curve
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
t Test for a Single Sample
• Comparison
distribution is the
distribution of means
– Figuring variance of
the distribution of
means
– Figuring standard
deviation of the
distribution of means
S
2
SM 
N
2
S M  S M2
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
t Test for a Single Sample
• Determine cutoff
sample score for
rejecting the null
hypothesis (using t
table)
• Figure sample mean’s
score on the t
distribution (t score)
M 
t
SM
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
t Test for Dependent Means
• Used to compare two sets of scores where there
are two scores for each person
– Repeated-measures
– Within-subjects
– Paired
• Compares mean difference score across pairs of
scores against a difference of 0 under the null
hypothesis.
• In other respects, t test for dependent means is just
like a single sample t test with a population mean
of 0
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Assumptions
• Assume that the population of individuals from
which the sample was taken is normally
distributed
• In practice, one seldom knows if a population is
normally distributed
– OK because many distributions in nature do
approximate a normal curve
– The t test is often still fairly accurate even when this
assumption is violated
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Effect Size
• Effect size for the t test for
dependent means is
– the mean of the difference
scores
– divided by the estimated SD
of the population of
individual difference scores
• Studies using the t test for
dependent means typically
have larger effect sizes
and more power than do
studies with participants
divided into two groups
1   2
d

• Effect size conventions
– Small = .20
– Medium = .50
– Large = .80
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall