Central Texas College Independent Measures T Statistic Key Concept Discussion

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Chapter 10
The t Test for Two
Independent Samples
PowerPoint Lecture Slides
Essentials of Statistics for the Behavioral Sciences
Tenth Edition
by Frederick J Gravetter, Larry B. Wallnau, Lori-Ann B. Forzano, and James E. Witnauer
Chapter 10 Learning Outcomes




Understand structure of research study appropriate
for independent-measures t hypothesis test
Test the difference between two populations or two
treatments using independent-measures t statistic
Evaluate the magnitude of the observed mean
difference (effect size) using Cohen’s d, r2, and/or a
confidence interval
Understand how to evaluate the assumptions
underlying this test and how to adjust calculations
when needed
Tools You Will Need



Sample variance (Chapter 4)
Standard error formulas (Chapter 7)
The t statistic (Chapter 9)



Distribution of t values
df for the t statistic
Estimated standard error
10-1 Introduction to the Independent
Measures Design (1 of 2)

Most research studies compare two (or more) sets
of data



Data from two completely different, independent
participant groups (an independent-measures
research design or between-subjects design)
Data from the same or related participant group(s)
(a within-subjects design or repeated-measures
research design)
The key concept is to explain the difference
between a one sample t test and an independent
samples t test.
Introduction to the Independent-Measures
Design (2 of 2)



Computational procedures are considerably
different for the two designs
Each design has different strengths and
weaknesses
Consequently, only between-subjects designs are
considered in this chapter; repeated-measures
research designs will be discussed in Chapter 11
Figure 10.1 The Structure of an IndependentMeasures Research Study
10-2 The Hypotheses and the IndependentMeasures t Statistic

Null hypothesis for independent-measures test (no
difference between the population means)

Alternative hypothesis for the independentmeasures test (there is a mean difference)
The Formulas for an Independent-Measures
Hypothesis Test


The basic structure of the t statistic
t = [(sample statistic) – (hypothesized population
parameter)] divided by the estimated standard error
The Estimated Standard Error


Measure of standard or average distance between
sample statistic (M1 – M2) and the population
parameter
How much difference is reasonable to expect
between two sample means if the
null hypothesis is true (Equation 10.1)?
Box 10.1 The Variability of Difference Scores


Why add sample measurement errors (squared
deviations from mean) but subtract sample means
to calculate a difference score?
The variability for the difference in scores is found
by adding the variability for each of the two
populations
Figure 10.2 Two Population Distributions
Pooled Variance

Equation 10.1 shows the standard error concept but
is unbiased only if the two samples are exactly the
same size ( n1 = n2)

Pooled variance (sp2) provides an unbiased basis
for calculating the standard error
Estimated Standard Error

The estimated standard error of M1 – M2 is then
calculated using the pooled variance estimate
The Final Formula and Degrees of Freedom


The independent-measures t statistic = sample
mean difference minus population mean difference,
divided by the estimated standard error
Degrees of freedom (df) for the t statistic is
df for first sample + df for second sample
Learning Check 1 (1 of 2)

Which combination of factors is most likely to produce a
significant value for an independent-measures t
statistic?
A. a small mean difference and small sample
variances
B. a large mean difference and large sample variances
C. a small mean difference and large sample variances
D. a large mean difference and small sample variances
Learning Check 1 – Answer (1 of 2)

Which combination of factors is most likely to produce a
significant value for an independent-measures t
statistic?
A. a small mean difference and small sample
variances
B. a large mean difference and large sample variances
C. a small mean difference and large sample variances
D. a large mean difference and small sample
variances
Learning Check 1 (2 of 2)

Decide if each of the following statements
is True or False.

T/F


If both samples have n = 10, the independentmeasures t statistic will have df = 19.
T/F

For an independent-measures t statistic, the
estimated standard error measures how much
difference is reasonable to expect between the
sample means if there is no treatment effect.
Learning Check 1 – Answers (2 of 2)

False


df = (n1 – 1) + (n2 – 1) = 9 + 9 = 18
True

This is an accurate interpretation
10-3 Hypothesis Tests with the IndependentMeasures t Statistic

Independent-measures hypothesis tests use the
same four steps as other hypothesis tests we’ve
discussed




State the hypotheses and select the alpha level
Locate the critical region
Obtain the data & compute the test statistic
Make a decision
Figure 10.3 The Critical Region for the IndependentMeasures Hypothesis Test in Example 10.2 (df = 14; α = .05)
Directional Hypotheses and One-Tailed Tests
• Use a directional test only when predicting a specific
direction of the difference is justified
• Locate critical region in the appropriate tail
• Report the use of a one-tailed test explicitly in the
research report, for example,
t (14) = –2.67, p ≤ .01, one-tailed
Assumptions Underlying the IndependentMeasures t Formula



The observations within each sample must be
independent
The two populations from which the samples are
selected must be normal
The two populations from which the samples are
selected must have equal variances (called
homogeneity of variance)
Hartley’s F-Max Test

Test for homogeneity of variance


Large value indicates a large difference between
the sample variances
Small value (near 1.00) indicates similar sample
variances and that the homogeneity assumption is
reasonable
Pooled Variance Alternative

If sample information suggests violation of
homogeneity of variance assumption


Calculate standard error, as in Equation 10.1
Adjust df for the t test as given below:
10-4 Effect Size and Confidence Intervals for the
Independent-Measures t

If the null hypothesis is rejected, the effect size
should be determined by using either

Cohen’s estimated d

or Percentage of variance explained
Confidence Intervals for Estimating μ1 – μ2


Sample mean difference M1 – M2 is used to estimate
the population mean difference
t equation is solved for the unknown parameter (μ1 – μ2)
Confidence Intervals and Hypothesis Tests




Estimation can provide an indication of the size of
the treatment effect
Estimation can provide an indication of the
significance of the effect
If the interval contains zero, then it is not a
significant effect
If the interval does not contain zero, the treatment
effect was significant
Figure 10.4 The 95% Confidence Interval for the
Population Mean Difference from Example 10.6
Reporting the Results of an IndependentMeasures t Test





Report whether the difference between the two
groups was significant
Report descriptive statistics (M and SD) for each
group
Report t statistic and df
Report p-value
Report CI immediately after t, for example, 95% CI
[0.782, 7.218]
10-5 The Role of Sample Variance and Sample
Size in the Independent-Measures t Test



The standard error is directly related to sample
variance
Because larger variance leads to a larger error, it
also leads to a smaller value of t (closer to zero)
and reduces the likelihood of finding a significant
result
Because larger samples produce smaller error,
larger samples lead to a larger value for the t
statistic
Figure 10.5 Two Sample Distributions
Representing Two Different Treatments
Figure 10.6 Two Sample Distributions
Representing Two Different Treatments
Learning Check 2 (1 of 2)

For an independent-measures research study, the
value of Cohen’s d or r2 helps to describe _____.
A. the risk of a Type I error
B. the risk of a Type II error
C. how much difference there is between the
two treatments
D. whether the difference between the two
treatments is likely to have occurred by
chance
Learning Check 2 – Answer (1 of 2)

For an independent-measures research study, the
value of Cohen’s d or r2 helps to describe _____.
A. the risk of a Type I error
B. the risk of a Type II error
C. how much difference there is between
the two treatments
D. whether the difference between the two
treatments is likely to have occurred by
chance
Learning Check 2 (2 of 2)

Decide if each of the following statements
is True or False.

T/F


The homogeneity assumption requires the two
sample variances to be equal
T/F

If a researcher reports that t(6) = 1.98, p > .05, then
H0 was rejected
Learning Check 2 – Answers (2 of 2)

False


The assumption requires equal population
variances but the test is valid if sample variances
are similar
False

H0 is rejected when p < .05, and t > the critical
value of t
SPSS Output for the Independent-Measures
Hypothesis Test for Example 10.2
Group Statistics
VAR00002
VAR00001
N
Mean
Std. Deviation
Std. Error Mean
1.00
8
8.0000
2.92770
1.03510
2.00
8
12.0000
3.07060
1.08562
Independent Sample test
Levene’s Test for
Equality of Variances:
F
VAR00001
Equal variances
assumed
Levene’s Test for
Equality of Variances:
Sig.
.000
1.000
Equal variances
not assumed
Test for
Equality of
Means: t
Test for Equality of
Means: df
−2.667
14
−2.667
13.968
Independent Sample test
Test for
Equality of
Means: Sig. (2tailed)
VAR00001
Test for
Equality of
Means: Means
Difference
Test for Equality
of Means: Std.
Error Difference
Test for Equality of
Means: 95%
Confidence Interval of
the Difference: Lower
Test for Equality of Means:
95% Confidence Interval of
the Difference: Upper
Equal variances
assumed
.018
−4.00000
1.50000
−7.21718
−.78282
Equal variances not
assumed
.018
−4.00000
1.50000
−7.21786
−.78214
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