Testing Relationships between Variables

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Testing Relationships between Variables
Statistics 111 - Lecture 17
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Administrative Notes
Administrative Notes

• Homework 5 due tomorrow
• Lecture on Wednesday will be review of entire
  course

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Final Exam

• Thursday from 1040-1210
• Itll be right here in this room
• Calculators are definitely needed!
• Single 8.5 x 11 cheat sheet (two-sided) allowed
• Ive put a sample final on the course website

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Outline

• Review of Regression coefficients
• Hypothesis Tests
• Confidence Intervals
• Examples

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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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Best values for Regression Parameters

• The best fit line has these values for the
  regression coefficients
• Also can estimate the average squared residual

Best estimate of slope ?
Best estimate of intercept ?
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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 versus Ha ? ? 0

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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 (ei) in each
  prediction
• We approximate the error by using the residual

Observed Yi
Predicted Yi ? ?Xi
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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!

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Example Education and Mortality
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Confidence Intervals for Coefficients

• JMP output also gives the information needed to
  make confidence intervals for slope and intercept
• 100C confidence interval for slope ?
• b /- tn-2 SE(b)
• The multiple t comes from a t distribution with
  n-2 degrees of freedom
• 100C 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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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 2.08.307 (-54.2,-21.0)
• Note that this interval does not contain zero!
• 95 confidence interval for intercept ?
• 1353 2.091.42 (1170,1536)

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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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Another Example Draft Lottery

• p-value lt 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 ?
• -.231.98.05 (-.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

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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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Education Example

• Is the positive linear association we see between
  GPA and IQ statistically significant?

p-value
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Education Example

• p-value lt 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 ?
• .1011.99.014 (.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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Next Class - Lecture 18

• Review of course material