The IndependentSamples t Test
Chapter 11
Independent Samples t-Test
> Used to compare two means in a
between-groups design (i.e., each
participant is in only one condition)
Distribution of Differences Between
Means
Hypothesis Tests & Distributions
Steps for Calculating Independent
Sample t Tests
> Step 1: Identify the populations,
distribution, and assumptions.
> Step 2: State the null and research
hypotheses.
> Step 3: Determine the characteristics of
the comparison distribution.
> Step 4: Determine critical values, or
cutoffs.
> Step 5: Calculate the test statistic.
> Step 6: Make a decision.
Step 1: Identify the populations,
distribution, and assumptions.
Population 1: People told they are drinking wine from a $10
bottle.
Population 2: People told they are drinking wine from a $90
bottle.
The distribution: a distribution of differences between means
(rather than a distribution of mean difference scores).
Assumptions: The participants were not randomly selected
so we must be cautious with respect to generalizing our
findings.
We do not know whether the population is normally
distributed.
Step 2: State the null and
research hypotheses.
Null hypothesis: On average, people drinking wine
they were told was from a $10 bottle give it the same
rating as people
drinking wine they were told was from a $90 bottle.
H0: μ1=μ2
Research hypothesis: On average, people drinking
wine they were told was from a $10 bottle give it a
different rating than people drinking wine they were
told was from a $90 bottle.
H1: μ1 ≠ μ2
Step 3: Determine the characteristics of
the comparison distribution.
Calculate the pooled variance and then the
standard deviation of the difference.
Formulae
s
2
X
(X  M )


2
Y
s
N 1
s
2
pooled
2
difference
s
2
s
(Y  M )


2
N 1
 df X  2  dfY  2
s X  
sY
 
 df total 
 df total 
2
MX
s
2
MY
sdifference  s
2
difference
Additional Formulae
( M X  M Y )  (  X  Y )
t
sdifference
M X  MY
t
sdifference
Step 4: Determine critical
values, or cutoffs.
Step 5. Calculate the test
statistic
( M X  M Y )  (  X  Y )
t
sdifference
M X  MY
t
sdifference
Step 6: Make a Decision.
Reporting the Statistics
> t(df) = tcalc, p < .05
• Use p > .05 if there is no difference
between means
• Use p < .05 if there is a difference between
means
> t(7) = -2.44, p < .05
Beyond Hypothesis Testing
> Just like z tests, single-sample t tests,
and paired-samples t tests, we can
calculated confidence intervals and
effect size for independent-samples
t tests
Steps for Calculating CIs
> Step 1. Draw a normal curve with the sample
difference between means in the center.
> Step 2. Indicate the bounds of the CI on either
end, writing the percentages under each
segment of the curve.
> Step 3. Look up the t values for lower and
upper ends of the CIs in the t table.
> Step 4. Convert the t values to raw
differences.
> Step 5. Check the answer.
A 95% Confidence Interval for
Differences Between Means, Part I
A 95% Confidence Interval for
Differences Between Means, Part II
A 95% Confidence Interval for
Differences Between Means, Part III
Effect Size
> Used to supplement hypothesis testing
> Cohen’s d:
( M X  M Y )  (  X  Y )
d
s pooled
Effect Size
Data Transformations
1. Transform a scale variable to an ordinal
variable.
2. Use a data transformation such as
square root transformation to “squeeze”
the data together to make it more normal.
> Remember that we need to apply any
kind of data transformation to every
observation in the data set.
Stop and Think
> When would you use a z test over a t
test?
> When would you use an independent
sample t test? Think of a specific study.
> When would you use a paired sample t
test? Think of a specific study.