Transcript Document 7567197

Correlation and the logic of
significance testing
Practice
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A researcher examines whether being in a room with
blue walls, green walls, red walls, or beige walls
influences aggressive behavior in a group of
adolescents (measured as the number of times the
adolescents shock an opponent in a simulated game).
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What is the independent variable?
Is the independent variable manipulated or not manipulated?
How many levels are there for the independent variable?
What is the dependent variable?
What is the scale of measurement for the dependent variable?
Practice
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A researcher examines whether being subliminally
exposed to various novel shapes will increase
participants’ liking of the shapes. Some participants
subliminally view shapes and others do not. Later, all
participants are given a list of shapes and are asked to
rank-order their top 5 choices.
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What is the independent variable?
Is the independent variable manipulated or not manipulated?
How many levels are there for the independent variable?
What is the dependent variable?
What is the scale of measurement for the dependent variable?
Practice
• Q: True or False? If False, explain why.
Generally, a small standard deviation
implies that the measurements are
clustered close to the mean.
Practice
• Q: If a constant were to be added to a set
of scores, the standard deviation would:
• a. remain the same.
• b. increase by the square root of that
constant.
• c. increase by the square of that constant.
• d. increase by the magnitude of that
constant. e. none of the above.
Practice
• Q: If the variance of a distribution is 9, the
standard deviation is:
• a. 3
• b. 6
• c. 9
• d. 81
• e. impossible to determine without
knowing n.
Practice
• Q: The following set of scores is obtained on a
test, X: 4, 6, 8, 9, 11, 13, 16, 24, 24, 24, 26. The
teacher computes all of the descriptive indices of
central tendency and variability on these data,
then discovers that an error was made, and one
of the 24's is actually a 17. Which of the
following indices will be changed from the
original computation?
• a. Median
b. Mode
• c. Range
d. Standard deviation
• e. None of the above
Practice
• Q: True or False? The standard deviation
of a group of scores is 0 when all the
scores are the same.
Correlation
• Correlation Coefficient
– A single number representing the degree of relation between two
variables.
– The value of a correlation coefficient can range from –1 to +1.
Relations between two variables
Computing the Correlation
Coefficient (Pearson R)
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Example: Computing Pearson R
X
Y
Zx
Zy
12.00
7.00
.17700
.52350
15.00
3.00
.81419
-1.27135
8.00
7.00
-.67259
.52350
4.00
4.00
-1.52218
-.82264
11.00
5.00
-.03540
-.37393
17.00
9.00
1.23899
1.42092
Z scores and cross-products
Zx
Zy
ZxZy
0.177
0.524
0.09
0.814
-1.271
-1.03
-0.673
0.5235
0.351
-1.522
-0.822
0.250
-0.035
-0.374
0.01
1.239
1.420
1.76
Computing the Correlation
Coefficient (Pearson R)
• r = 1.73/6
• r = .29
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1.73
Significance
• How correlated do our samples have to be
to conclude that the populations they
came from are correlated
• In other words – do we have a significant
correlation?
Logic of significance testing
• Null hypothesis – populations are not
correlated – correlation in the sample is
just due to chance
• Alternative hypothesis – populations are
correlated
Looking at some analogies
• Criminal justice
– Law and Order example
• Gambling
– Fair coin example
• Medicine
– Irregular cell example
• Guessing game
– Are numbers in envelope the same or
different?
Statistical Evidence
• Tables of significance
• Sample of correlation
table
DF
Probability, p
0.05
0.01
0.001
1
0.997
1.000
1.000
2
0.950
0.990
0.999
3
0.878
0.959
0.991
4
0.811
0.917
0.974
5
0.755
0.875
0.951
6
0.707
0.834
0.925
7
0.666
0.798
0.898
8
0.632
0.765
0.872
9
0.602
0.735
0.847
10
0.576
0.708
0.823
Possible errors
No correlation
Correlation
Conclude
Not
Significant
Correct reject
Type 2 error – A
miss
Conclude
Significant
Type I error – False Hit
alarm
Practice
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Suppose a researcher comes up with a new
drug that in fact cures AIDS. She assigns
some AIDS patients to receive a placebo, and
others to receive the new drug. The null
hypothesis is that the drug will have no effect.
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In this case, what would be the Type I error?
In this case, what would be the Type II error?
Which do you think is the more costly error? Why?
Practice
•
A manufacturing company is making smoke
detectors. They want to conduct a study to
determine how well the smoke detectors work.
The null hypothesis in this study is that the
smoke detectors will not work.
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In this case, what would be the Type I error?
In this case, what would be the Type II error?
Which do you think is the more costly error? Why?
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In one famous study (Rosenhan, 1973), eight healthy
volunteers (“pseudopatients”) presented themselves to
psychiatric hospitals. They told the admissions officers
that they had been hearing voices.
All pseudopatients were admitted to the hospitals,
where they ceased to pretend they were hearing voices
(in other words, they behaved ‘normally’). None of the
pseudopatients were detected as sane.
In this study, did the hospital staff commit a Type I or Type II error
in calling sane patients insane?
Which would be the more costly error? Why?
If we conclude significance what
does it mean
• Scores in sample were so highly
correlated that it is unlikely that this was
due to chance – thus we conclude that
populations are correlated.
Significance testing in a simple
experiment
• Two conditions compared
• Are differences between the means of the
samples so different that it is likely that the
populations that they came from were
different
• In other words – Is there a significant
difference between the groups
Significance testing for differences
between means
• Null hypothesis: There is no difference
between the populations (any differences
you find in the samples are due to chance)
• Alternative hypothesis: There is a
difference between the populations (any
difference you find in the samples reflect
the differences between the populations)
Test for differences between two
means
• The t-test
• Looks at differences between the means
in relation to the variability of the scores
Next time
• Different types of t-tests
– One tail v Two tail
• Understanding the logic of the t-test
• Matching the t-test to the experimental
method