Inferences About The Pearson Correlation Coefficient

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Inferences About The Pearson Correlation
Coefficient
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STUDENTS Y(GPA) X(SAT)
A 1.6 400 -0.97 -145.80 141.43
B 2.0 350 -0.57 -195.80 111.61
C 2.2 500 -0.37 -45.80 16.95
D 2.8 400 0.23 -145.80 -33.53
E 2.8 450 0.23 -95.80 -22.03
F 2.6 550 0.03 4.20 0.13
G 3.2 550 0.63 4.20 2.65
H 2.0 600 -0.57 54.20 -30.89
I 2.4 650 -0.17 104.20 -17.71
J 3.4 650 0.83 104.20 86.49
K 2.8 700 0.23 154.20 35.47
L 3.0 750 0.43 204.20 87.81
Sum 30.80 6550.0 378.33
Mean 2.57 545.80
S.D. 0.54 128.73
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Calculation of Covariance Correlation
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Population of visual acuity and neck size
scores ?0
Sample 1
Sample 2
Sample 3
Etc
r -0.8
r .15
r .02
Relative Frequency
0 µr
r
The development of a sampling distribution of
sample v
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Steps in Test of Hypothesis

1. Determine the appropriate test
2. Establish the level of significancea
3. Determine whether to use a one tail or two tail
   test
4. Calculate the test statistic
5. Determine the degree of freedom
6. Compare computed test statistic against a
   tabled/critical value

Same as Before
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1. Determine the Appropriate Test

• Check assumptions
• Both independent and dependent variable (X,Y) are
  measured on an interval or ratio level.
• Pearsons r is suitable for detecting linear
  relationships between two variables and not
  appropriate as an index of curvilinear
  relationships.
• The variables are bivariate normal (scores for
  variable X are normally distributed for each
  value of variable Y, and vice versa)
• Scores must be homoscedastic (for each value of
  X, the variability of the Y scores must be about
  the same)
• Pearsons r is robust with respect to the last
  two specially when sample size is large

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2. Establish Level of Significance

• a is a predetermined value
• The convention
• a .05
• a .01
• a .001

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3. Determine Whether to Use a One or Two Tailed
Test

• H0 ?XY 0
• Ha ?XY ? 0
• Ha ?XY gt or lt 0

Two Tailed Test if no direction is specified
One Tailed Test if direction is specified
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4. Calculating Test Statistics
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5. Determine Degrees of Freedom

• For Pearsons r df N 2

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6. Compare the Computed Test Statistic Against a
Tabled Value

• a .05
• Identify the Region (s) of Rejection.
• Look up ta corresponding to degrees of freedom

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Example of Correlations Between SAT and GPA scores

• Formulate the Statistical Hypotheses.
• Ho ?XY 0 Ha ?XY ? 0
• a 0.05
• Collect a sample of data, n 12

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Data
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Calculation of Difference of Y and mean of Y
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Calculation of Difference of X and Mean of X
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Calculation of Product of Differences
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Covariance Correlation
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Calculate t-statistics
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Check Significance

• Identify the Region (s) of Rejection.
• ta 2.228
• Make Statistical Decision and Form Conclusion.
• tc lt ta Fail to reject Ho
• p-value 0.095 gt a 0.05 Fail to reject Ho
• Or use Table B-6 rc 0.50 lt ra .576 Fail to
  reject Ho

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Practical Significance in Pearson r

• Judge the practical significance or the magnitude
  of r within the context of what you would expect
  to find, based on reason and prior studies.
• The magnitude of r is expressed in terms of r2 or
  the coefficient of determination.
• In our example, r2 is .50 2 .25 (The proportion
  of variance that is shared by the two variables).

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Intuitions about Percent of Variance Explained
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Sample Size in Pearson r

• To estimate the minimum sample size needed in r,
  you need to do the power analysis. For example,
  Given the
• a .05, effect size (population r or ?)
  0.20, and a power of .80, 197 subjects would be
  needed. (Refer to Table 9-1).
• Note ? .10 (small), ?.30 (medium), ? .50
  (large)

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Magnitude of Correlations

• ? .10 (small)
• ? .30 (medium)
• ? .50 (large)

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Factors Influencing the Pearson r

• Linearity. To the extent that a bivariate
  distribution departs from normality, correlation
  will be lower.
• Outliers. Discrepant data points affect the
  magnitude of the correlation.
• Restriction of Range. Restricted variation in
  either Y or X will result in a lower correlation.
• Unreliable Measures will results in a lower
  correlation.

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Take Home Lesson

• How to calculate correlation and test if it is
  different from a constant