Statistics 111 - Lecture 17
Testing Relationships
between Variables
July 1, 2008
Lecture 17 - Regression Testing
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Administrative Notes
• Homework 5 due tomorrow
• Lecture on Wednesday will be review of
entire course
July 1, 2008
Lecture 17 - Regression Testing
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Final Exam
• Thursday from 10:40-12:10
• It’ll be right here in this room
• Calculators are definitely needed!
• Single 8.5 x 11 cheat sheet (two-sided)
allowed
• I’ve put a sample final on the course website
July 1, 2008
Lecture 17 - Regression Testing
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Outline
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Review of Regression coefficients
Hypothesis Tests
Confidence Intervals
Examples
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Lecture 17 - Regression Testing
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Two Continuous Variables
• Visually summarize the relationship between two
continuous variables with a scatterplot
• Numerically, we focus on best fit line (regression)
Education and Mortality
Draft Order and Birthday
Mortality = 1353.16 - 37.62 · Education
Draft Order = 224.9 - 0.226 · Birthday
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Lecture 17 - Regression Testing
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Best values for Regression Parameters
• The best fit line has these values for the regression
coefficients:
Best estimate of slope 
Best estimate of intercept

• Also can estimate the average squared residual:
June 30, 2008
Stat 111 - Lecture 16 - Regression
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Significance of Regression Line
• Does the regression line show a significant linear
relationship between the two variables?
• If there is not a linear relationship, then we would
expect zero correlation (r = 0)
• So the slope b should also be zero
• Therefore, our test for a significant relationship will
focus on testing whether our slope  is significantly
different from zero
H0 :  = 0
July 1, 2008
versus
Ha :   0
Lecture 17 - Regression Testing
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Linear Regression
• Best fit line is called Simple Linear Regression
Model:
• Coefficients:is the intercept and  is the slope
• Other common notation: 0 for intercept, 1 for slope
• Our Y variable is a linear function of the X variable but
we allow for error (εi) in each prediction
• We approximate the error by using the residual
Observed Yi
Predicted Yi = + Xi
June 30, 2008
Stat 111 - Lecture 16 - Regression
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Test Statistic for Slope
• Our test statistic for the slope is similar in form to all
the test statistics we have seen so far:
• The standard error of the slope SE(b) has a
complicated formula that requires some matrix
algebra to calculate
• We will not be doing this calculation manually
because the JMP software does this calculation
for us!
June 30, 2008
Stat 111 - Lecture 16 - Regression
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Example: Education and Mortality
June 30, 2008
Stat 111 - Lecture 16 - Regression
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Confidence Intervals for Coefficients
• JMP output also gives the information needed to
make confidence intervals for slope and intercept
• 100·C % confidence interval for slope  :
b +/- tn-2* SE(b)
• The multiple t* comes from a t distribution with n-2
degrees of freedom
• 100·C % confidence interval for intercept  :
a +/- tn-2* SE(a)
• Usually, we are less interested in intercept  but it
might be needed in some situations
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Lecture 17 - Regression Testing
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Confidence Intervals for Example
• We have n = 60, so our multiple t* comes from a t
distribution with d.f. = 58. For a 95% C.I., t* = 2.00
• 95 % confidence interval for slope  :
-37.6 ± 2.0*8.307 = (-54.2,-21.0)
Note that this interval does not contain zero!
• 95 % confidence interval for intercept  :
1353± 2.0*91.42 = (1170,1536)
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Lecture 17 - Regression Testing
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Another Example: Draft Lottery
• Is the negative linear association we see between
birthday and draft order statistically significant?
p-value
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Lecture 17 - Regression Testing
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Another Example: Draft Lottery
• p-value < 0.0001 so we reject null hypothesis and
conclude that there is a statistically significant
linear relationship between birthday and draft order
• Statistical evidence that the randomization was not done
properly!
• 95 % confidence interval for slope  :
-.23±1.98*.05 = (-.33,-.13)
• Multiple t* = 1.98 from t distribution with n-2 = 363 d.f.
• Confidence interval does not contain zero, which we
expected from our hypothesis test
July 1, 2008
Lecture 17 - Regression Testing
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Education Example
• Dataset of 78 seventh-graders: relationship
between IQ and GPA
• Clear positive association between IQ and
grade point average
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Lecture 17 - Regression Testing
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Education Example
• Is the positive linear association we see between
GPA and IQ statistically significant?
p-value
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Lecture 17 - Regression Testing
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Education Example
• p-value < 0.0001 so we reject null hypothesis and
conclude that there is a statistically significant
positive relationship between IQ and GPA
• 95 % confidence interval for slope  :
.101±1.99*.014 = (.073,.129)
• Multiple t* = 1.99 from t distribution with n-2 = 76 d.f.
• Confidence interval does not contain zero, which we
expected from our hypothesis test
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Lecture 17 - Regression Testing
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Next Class - Lecture 18
• Review of course material
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Lecture 17 - Regression Testing
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