Download Chapter 9 - Bakersfield College

Chapter 9
HYPOTHESIS TESTING USING THE
TWO-SAMPLE t-TEST
Going Forward
Your goals in this chapter are to learn:
• The logic of a two-sample experiment
• The difference between independent samples
and related samples
• When and how to perform the independentsamples t-tests
• When and how to perform the relatedsamples t-test
• What effect size is and how it is measured
2
using Cohen’s d or rpb
Understanding the
Two-Sample Experiment
Two-Sample Experiment
• Participants’ scores are measured under two
conditions of the independent variable
• Condition 1 produces sample mean X 1
representing 1
• Condition 2 produces sample mean X 2
representing  2
Two-Sample t-Test
• The parametric statistical procedure for
determining whether the results of a
two-sample experiment are significant is
the two-sample t-test
• The two versions of the two-sample
t-test are
– The independent-samples t-test
– The related-samples t-test
Relationship in the Population in a
Two-sample Experiment
The Independent Samples t-Test
Independent Samples t-Test
• The parametric procedure used for testing two
sample means from independent samples
• Independent samples result when we
randomly select participants for a condition
without regard to who else has been selected
for either condition
Assumptions of the Independent
Samples t-Test
• The dependent scores are normally
distributed interval or ratio scores.
• The populations have homogeneous
variance. Homogeneity of variance means
the variance of the populations being
2
represented ( X ) are equal.
Statistical Hypotheses
• For a two-tailed test, the statistical hypotheses
are
H 0 : 1   2  0
H a : 1   2  0
• H0 implies both samples represent the same
population of scores
• Ha implies the means from our conditions
each represent a different population of
scores
Sampling Distribution
The sampling distribution of differences
between the means is the distribution of all
possible differences between two means when
both samples are drawn from the one raw score
population that H0 says we are representing.
Performing the Independent
Samples t-Test
1. Compute the mean and estimated population
variance for each condition
Remember: The formula for the estimated variance
in each condition is
( X )
X 
2
n
sX 
n 1
2
2
Performing the Independent
Samples t-Test
2. Compute the pooled variance using the
formula
s
2
pool
(n1  1) s  (n2  1) s

(n1  1)  (n2  1)
2
1
2
2
Performing the Independent
Samples t-Test
3. Compute the standard error of the
difference. This is the standard deviation of
the sampling distribution of differences
between means. The formula is
s X1  X 2  ( s
2
pool
1
1
)  
 n1 n2 
Performing the Independent
Samples t-Test
4. Compute tobt for two independent samples
using the formula
tobt
( X 1  X 2 )  ( 1   2 )

s X1  X 2
One-Tailed Tests
The statistical hypotheses for a one-tailed test of
independent samples are
H 0 : 1   2  0
H a : 1   2  0
If 1 is expected to
be larger than 2
OR
H 0 : 1   2  0
H a : 1   2  0
If 2 is expected to
be larger than 1
One-Tailed Tests
Conduct one-tailed tests only when you can
confidently predict the direction the dependent
scores will change.
One-Tailed Tests
1. Decide which X and corresponding  is
expected to be larger
2. Arbitrarily decide which condition to subtract
from the other
3. Decide whether the difference will be
positive or negative
4. Create Ha and H0 to match this prediction
5. Locate the region of rejection
6. Complete the t-test as described previously
Critical Values
Critical values for the independent samples ttest (tcrit) are determined based on
• degrees of freedom df = (n1 – 1) + (n2 – 1),
• the selected a, and
• whether a one-tailed or two-tailed test
is used
Interpreting the
Independent-Samples t-Test
• In a two-tailed t-test of independent samples,
reject H0 if tobt is greater than (beyond) +tcrit or
if tobt is less than (beyond) –tcrit
• Otherwise, fail to reject H0
The Related Samples t-Test
Related-Samples t-Test
The related-samples t-test is used when we have
two sample means from two related samples
• Related samples occur when we pair each
score in one sample with a particular score in
the other sample
• Two types of research designs producing
related samples are the matched-samples
design and the repeated-measures design
Matched-Samples Design
• The researcher matches each participant in
one condition with a particular participant in
the other condition
• We do this so we have more comparable
samples
Repeated-Measures Design
• Each participant is tested under both
conditions of the independent variable
• That is, each participant is measured under
condition 1 and again under condition 2
Transforming the Raw Scores
• In a related samples t-test, the raw scores are
transformed by finding each difference score
• The difference score is the difference between
the two raw scores in a pair
• The symbol for a difference score is D
Statistical Hypotheses
The statistical hypotheses for a two-tailed
related-samples t-test are
H0 : D  0
Ha : D  0
Sampling Distribution
The sampling distribution of mean differences
shows all possible values of the population
mean of the difference scores ( D ) that occur
when samples are drawn from the population of
difference scores that H0 says we are
representing.
Performing the
Related-Samples t-Test
1. Compute the estimated variance of the
2
difference scores ( s D ) using the formula
( D )
D 
N
sD2 
N 1
2
2
where N equals the number of difference
scores
Performing the
Related-Samples t-Test
2. Compute the standard error of the mean
difference ( s D ) using the formula
sD 
sD2
N
Performing the
Related-Samples t-Test
3. Find tobt using the formula
tobt
D  D

sD
One-Tailed Tests
The statistical hypotheses for a one-tailed t-test
of related samples are
H0 : D  0
H0 : D  0
Ha : D  0
Ha : D  0
If we expect the
difference to be
larger than 0
If we expect the
difference to be
less than 0
Critical Values
The critical value (tcrit) is determined based on
• degrees of freedom df = N – 1 where N is the
number of difference scores
• the selected a, and
• whether a one-tailed or two-tailed test is used
Interpreting the
Related-Samples t-Test
• In a two-tailed test of related samples, reject
H0 if tobt is greater than (beyond) +tcrit or if tobt
is less than (beyond) –tcrit
• Otherwise, fail to reject H0
Describing Effect Size
Effect Size
• Effect size indicates the amount of influence
changing the conditions of the independent
variable had on dependent scores
• The larger the effect size, the more
scientifically important the independent
variable is
Computing Effect Size
Cohen’s d is used to compute effect size
Independent Samples
t-Test
d
X1  X2
2
s pool
Related Samples
t-test
d
D
2
sD
Interpreting Effect Size
We interpret the Cohen’s d using a small,
medium, or large effect size classification
• d = 0.2 is a small effect
• d = 0.5 is a medium effect
• d = 0.8 is a large effect
Proportion of Variance
Accounted For
• The proportion of variance accounted for is
the proportion of the differences in scores
that can be attributed to changing the
conditions in the independent variable
• We use the formula for the squared pointbiserial correlation coefficient
2
(
t
)
2
obt
rpb 
(t obt ) 2  df
Example 1
Using the following data set, conduct an
independent-samples t-test. Use a = 0.05 and a
two-tailed test.
Sample 1
Sample 2
14
14
13
15
11
15
13
10
12
13
14
13
14
15
17
14
14
15
Example 1
X 1  13.556
X 2  13.778
s 2pool
s1  1.944
s2  1.302
n1  9
n2  9
(n1  1) s12  (n2  1) s22

(n1  1)  (n2  1)
(9 1)3.779  (9 1)1.695

(9 1)  (9 1)
 2.737
Example 1
The standard error of the difference is
s X1  X 2  ( s
2
pool
1
1
)  
 n1 n2 
1 1
 (2.737)  
9 9
 (2.737)(0.222)  0.780
Example 1
tobt
( X 1  X 2 )  ( 1   2 )

s X1  X 2
(13.556 13.778)  0

0.780
 0.222

  0.285
0.780
Example 1
• tcrit for df = (9 – 1) + (9 – 1) = 16 with a = .05
and a two-tailed test is 2.120.
• Reject H0 if tobt is greater than +2.120 or if tobt
is less than –2.120.
• Because tobt of – 0.285 is not beyond the –tcrit
of –2.120, it does not lie within the rejection
region. We fail to reject H0.
Example 2
Using the following data set, conduct a related-samples
t-test. Use a = 0.05 and a two-tailed test.
Sample 1
Sample 2
14
14
13
15
16
15
13
10
12
16
14
13
14
15
17
18
17
19
Example 2
First, we find the differences between the
matched scores
Sample 1
Sample 2
Differences
14
14
13
15
16
15
-1
-2
-2
13
10
12
16
14
13
-3
-4
-1
14
15
17
18
17
19
-4
-2
-2
Example 2
tobt 
D  D
2
D
s
N
 2.333  0  2.333


  6.260
0.373
1.25
9
Example 2
• Using a = 0.05 and df = 8, tcrit = 2.306.
• Reject H0 if tobt is greater than +2.306 or if tobt
is less than –2.306.
• Because tobt of –6.260 is beyond the –tcrit
value of –2.306, it lies within the rejection
region. We reject H0.
Example 2
Effect size
d

D
2
sD
 2.333
1.25
  2.087
Example 2
Proportion of variance
accounted for
2
(
t
)
2
obt
rpb 
2
(tobt )  df
 6.26

 6.26 2  8
39.188

47.188
 0.830
2