Topics for Today
Testing for a significant correlation
Response and Explanatory Variables
Stat203
Fall2011 – Week 10, Lecture 2
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Hypothesis Tests for Correlations
A ___________________, r, is just another _________
(like a mean, or a median, or a standard deviation) which
can be measured on a sample, and can vary from sample
to sample.
So … the r we measure on any one sample gives us only a
guess at the true correlation ρ … and in the same way as
the t-test gave us a measure of how far away our sample
mean was from a hypothesized mean µ0, we can do the
same for a hypothesized ___________ ρ0 using our
______ correlation.
Note, though, we’re almost always only concerned with
whether the true correlation is different than zero.
Stat203
Fall2011 – Week 10, Lecture 2
Page 2 of 19
Review from last lecture:
The ________ deviation (standard error) of r:
1- r 2
seˆ(r) =
n -2
and the ____ statistic (it’s a t-test!)
r-0
r
t=
=
2
se(r)
1- r
Stat203
Fall2011 – Week 10, Lecture 2
n -2
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but first … another assumption
As with most t-tests, there’s an assumption we have to
check first.
Both of the variables in our correlation must be normally
distributed or ______ in order for us to use the t-test.
Checking normality in small samples is extremely difficult …
just make sure histograms of both variables aren’t ________
skewed or _______.
Stat203
Fall2011 – Week 10, Lecture 2
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Testing Correlation Example 1
Is wine good for your heart? There is some evidence that
drinking moderate amounts of wine helps prevent heart
attacks. The estimated yearly amounts of alcohol from wine
per person and the heart attack death rate were calculated
for 19 randomly selected countries. Is there any evidence
of a relationship?
First, scatterplot, then statistic, then hypothesis and
conclusion.
[source: MH Criqui, University of California, San Diego, reported in the New York
Times, December 28, 1994]
Stat203
Fall2011 – Week 10, Lecture 2
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Research Question:
Individual:
Population:
Variables:
Parameters:
Statistical Hypotheses:
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Fall2011 – Week 10, Lecture 2
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Pearson’s correlation:
Test-statistic (by hand):
p-value:
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Fall2011 – Week 10, Lecture 2
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…and from SPSS:
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Fall2011 – Week 10, Lecture 2
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Conclusion:
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Fall2011 – Week 10, Lecture 2
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Example 2 (C10Q15, pg 370): Is there a correlation
between poverty and rates of teen pregnancy in 8 US
states?
Research Question:
Individuals:
Population:
Variables:
Parameters:
Stat203
Fall2011 – Week 10, Lecture 2
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Statistical Hypotheses:
Pearson Correlation:
p-value:
Conclusion:
Stat203
Fall2011 – Week 10, Lecture 2
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Example: Age and TV Watching
A researcher thinks that there is a relationship between
someone’s age and the number of hours per day they spend
watching TV.
Research Question:
Individuals:
Population:
Variables:
Stat203
Fall2011 – Week 10, Lecture 2
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Parameters:
Statistical Hypotheses:
Pearson Correlation:
p-value:
Conclusion:
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Fall2011 – Week 10, Lecture 2
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But is this the right conclusion?
Always check the scatterplot.
Stat203
Fall2011 – Week 10, Lecture 2
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Response & Explanatory Variables
Response variable (Y)
 Sometimes called the dependent variable.
 Measures the outcome of a study.
Explanatory variable (X)
 Sometimes called the independent variable, a
covariate, or a predictor.
 Explains or influences changes in the response
variable.
Note: When we don’t always set the values of these
variables but often just observe both. In fact, sometimes
there may not be explanatory or response variables. It
depends on how we plan to use the data.
Stat203
Fall2011 – Week 10, Lecture 2
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Variables, Causality
Alcohol has many effects on the body. One effect is a drop
in body temperature. To study this effect, researchers gave
several different amounts of alcohol to mice, then
measured the body temperature 15 minutes after
administering the alcohol.
Alcohol actually causes a change in body.
Alcohol dose = Explanatory Variable
Body Temperature = Response Variable
Stat203
Fall2011 – Week 10, Lecture 2
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Stat 203 scores for Assignment #1 and scores for Quiz #1
are available for 214 students. We are going to predict the
quiz score knowing the assignment score.
Assignment Score does not effect the Quiz score.
Assignment score = Explanatory Variable
Quiz Temperature = Response Variable
Just because the scores are related, this is not necessarily
a cause and effect relationship … but we can still use one
score to predict another.
Stat203
Fall2011 – Week 10, Lecture 2
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For investigating the relationship between two variables,
identifying a response and explanatory variable is
unnecessary (recall correlation analyses).
However, if we are interested in actually predicting the
value of one variable for an individual, we must identify one
variable as explanatory and one as the response.
It is important to remember that calling one variable the
explanatory variable and the other the response doesn’t
necessarily mean that changes in one variable cause
changes in the other variable.
Stat203
Fall2011 – Week 10, Lecture 2
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New Topics Covered Today
Testing for a Significant Correlation
 Can use a form of t-test
 No definition of response or explanatory variable is necessary
 Always check the scatterplot before making a conclusion!
Response and Explanatory Variables
 Explanatory variables predict responses
 No CAUSALITY!
Reading:
Start Chapter 11
Stat203
Fall2011 – Week 10, Lecture 2
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