Overview

1
Overview

• Two paired samples Within-Subject Designs
• -Hypothesis test
• -Confidence Interval
• -Effect Size
• Two independent samples Between-Subject Designs
• Hypothesis test
• Confidence interval
• Effect Size

2
Comparing Two Populations
Until this point, all the inferential statistics
we have considered involve using one sample as
the basis for drawing conclusion about one
population. Although these single sample
techniques are used occasionally in real
research, most research studies aim to compare of
two (or more) sets of data in order to make
inferences about the differences between two (or
more) populations. What do we do when our
research question concerns a mean difference
between two sets of data?
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Two kinds of studies

• There are two general research strategies that
  can be used to obtain the two sets of data to be
  compared
• The two sets of data could come from two
  independent populations (e.g. women and men, or
  students from section A and from section B)
• The two sets of data could come from related
  populations (e.g. before treatment and after
  treatment)

4
Part I

• Two paired samples Within-Subject Designs
• -Hypothesis test
• -Confidence Interval
• -Effect Size

5
Paired T-Test for Within-Subjects Designs
Our hypotheses Ho ?D 0 HA ?D ? 0
To test the null hypothesis, well again compute
a t statistic and look it up in the t table.
6
Steps for Calculating a Test Statistic
7
Confidence Intervals for Paired Samples
8
Effect Size for Dependent Samples
One Sample d
Paired Samples d
9
Exercise
In Everitts study (1994), 17 girls being treated
for anorexia were weighed before and after
treatment. Difference scores were calculated for
each participant.
Test the null hypothesis that there was no change
in weight. Compute a 95 confidence interval for
the mean difference. Calculate the effect size
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Exercise
T-test
11
Exercise
Confidence Interval
12
Exercise
Effect Size
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Part II

• Two independent samples Between-Subject Designs
• -Hypothesis test
• -Confidence Interval
• -Effect Size

14
T-Test for Independent Samples
The goal of a between-subjects research study is
to evaluate the mean difference between two
populations (or between two treatment conditions).
We cant compute difference scores, so
Ho ?1 ?2
HA ?1 ? ?2
15
T-Test for Independent Samples
We can re-write these hypotheses as follows
Ho ?1 - ?2 0 HA ?1 - ?2 ? 0
To test the null hypothesis, well again compute
a t statistic and look it up in the t table.
16
T-Test for Independent Samples
17
T-Test for Independent Samples
Standard Error for a Difference in Means
18
T-Test for Independent Samples
Standard Error for a Difference in Means
Each of the two sample means represents its own
population mean, but in each case there is some
error. The amount of error associated with each
sample mean can be measured by computing the
standard errors. To calculate the total amount of
error involved in using two sample means to
approximate two population means, we will find
the error from each sample separately and then
add the two errors together.
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T-Test for Independent Samples
Standard Error for a Difference in Means
But This formula only works when n1 n2. When
the two samples are different sizes, this formula
is biased. This comes from the fact that the
formula above treats the two sample variances
equally. But we know that the statistics
obtained from large samples are better estimates,
so we need to give larger sample more weight in
our estimated standard error.
20
T-Test for Independent Samples
Standard Error for a Difference in Means
We are going to change the formula slightly so
that we use the pooled sample variance instead of
the individual sample variances.
This pooled variance is going to be a weighted
estimate of the variance derived from the two
samples.
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Steps for Calculating a Test Statistic
22
Steps for Calculating a Test Statistic
23
Illustration
A developmental psychologist would like to
examine the difference in verbal skills for
8-year-old boys versus 8-year-old girls. A
sample of 10 boys and 10 girls is obtained, and
each child is given a standardized verbal
abilities test. The data for this experiment are
as follows
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Illustration
STEP 1 get mean difference
25
Illustration
STEP 2 Compute Pooled Variance
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Illustration
STEP 3 Compute Standard Error
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Illustration
STEP 4 Compute T statistic and df
d.f. (n1 - 1) (n2 - 1) (10-1) (10-1) 18
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Illustration
STEP 5 Use table E.6
T 3 with 18 degrees of freedom
For alpha .01, critical value of t is 2.878 Our
T is more extreme, so we reject the null There is
a significant difference between boys and girls
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T-Test for Independent Samples
Sample Data Hypothesized Population Parameter Sample Variance Estimated Standard Error t-statistic
Single sample t-statistic
Independent samples t-statistic
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Confidence Intervals for Independent Samples
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Effect Size for Independent Samples
One Sample d
Independent Samples d
32
Exercise
Subjects are asked to memorize 40 noun pairs.
Ten subjects are given a heuristic to help them
memorize the list, the remaining ten subjects
serve as the control and are given no help. The
ten experimental subjects have a X-bar 21 and a
SS 100. The ten control subjects have a X-bar
19 and a SS 120.
Test the hypothesis that the experimental group
differs from the control group. Give a 95
confidence interval for the difference between
groups Give the effect size
33
Exercise
T-test
34
Exercise
T-test
d.f. (n1 - 1) (n2 - 1) (10-1) (10-1) 18
35
Exercise
Confidence Interval
36
Exercise
Effect Size
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Summary
Hypothesis Tests Confidence Intervals Effect Sizes
1 Sample 2 Paired Samples 2 Independent Samples
38
Review
39
Sample Data Hypothesized Population Parameter Sample Variance Estimated Standard Error t-statistic
One sample t-statistic
Paired samples t-statistic
Independent samples t-statistic
40
Confidence Intervals
41
Effect Sizes
One Sample d
Paired Samples d
Independent Samples d