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9.1 day 2 Correlation
April 20, 2017
© 2012 Pearson Education, Inc.
All rights reserved.
Section 9.1 Objectives
© 2012 Pearson Education, Inc. All rights reserved.
Introduce linear correlation, independent and dependent variables, and the types of correlation
Find a correlation coefficient (r)
Test a population correlation coefficient ρ
using a table
Perform a hypothesis test
Distinguish between correlation and causation
Using a Table to Test a Population Correlation Coefficient ρ
We need to determine whether there is enough evidence to decide that the population correlation coefficient ρ is significant at a specified level of significance.
Use Table Page A26
If |r| > ρ ,
there is enough evidence to decide that the correlation coefficient ρ is significant.
In Words In Symbols
Specify the level of significance.
Find the critical value.
Use Table Page A26.
Decide if the correlation is significant.
Interpret the decision in the context of the original claim.
If |r| > ρ the correlation is significant. Otherwise, not significant relationship between the two data identified
Example 6: Using a Table to Test a Population Correlation Coefficient ρ
Using the Old Faithful data, you used 25 pairs of data to find r ≈ 0.979.
Is the correlation coefficient significant?
Use α = 0.05. (table A26)
Solution: Using a Table to Test a Population Correlation Coefficient ρ
n = 25,α = 0.05
|r| ≈ 0.979 > 0.396
There is enough evidence at the 5% level of significance to conclude that there is a significant linear correlation between the duration of Old Faithful’s eruptions and the time between eruptions.
Hypothesis Testing for a Population Correlation Coefficient ρ
A hypothesis test can also be used
A hypothesis test can be one-tailed or two-tailed.
We are only using 2-tailed test.
H0: ρ = 0 (not significant correlation) *Ha: ρ ≠ 0 (significant correlation)
Using the t-Test for ρ
State the null and alternative hypothesis, and the level of significance.
Identify the degrees of freedom. (one loss for each variable)
Determine the critical value(s) and rejection region(s).
State H0 and Ha.
d.f. = n – 2*
Use Table Page A18.
4. Find the standardized test statistic.
5. Make a decision to reject or fail to reject the null hypothesis.
6.Interpret the decision in the context of the original claim.
If t is in the rejection region, reject H0. Otherwise FTR H0.
Example 7: t-Test for a Correlation Coefficient
(trillions of $), x
CO2 emission (millions of metric tons), y
Previously you calculated r ≈ 0.882.
Test the significance of this correlation coefficient.
Use α = 0.05.
ρ = 0
ρ ≠ 0
10 – 2* = 8
Solution: t-Test for a Correlation Coefficient
1) Reject H0
At the 5% level of significance, there is enough evidence to conclude that there is a significant linear correlation between gross domestic products and carbon dioxide emissions.
Correlation and Causation
Strong correlation does not imply a cause-and-effect relationship between the variables.
If there is a significant correlation between two variables, you should consider:
Is there a direct cause-and-effect relationship between the variables? (old faithful)
Does x cause y?
Is there a reverse cause-and-effect relationship between the variables? (old faithful)
Does y cause x?
Is it possible that the relationship between the variables can be caused by a third variable or by a combination of several other variables?
Ex: ice cream sales and crime in July
Is it possible that the relationship between two variables may be a coincidence?
Ex: types of animal species in a region and #2 – car owners in that region.
9.1 day 2 Assignment:
Larson/Farber 5th ed.
#7, 8, 29, 30, 33, 34
cision in the context of the original claim.
Solution: t-Test for a C