Stat 155, Section 2, Last Time

1
Stat 155, Section 2, Last Time

• Reviewed Excel Computation of
• Time Plots (i.e. Time Series)
• Histograms
• Modelling Distributions Densities (Areas)
• Normal Density Curve (very useful model)
• Fitting Normal Densities
• (using mean and s.d.)

2
Reading In Textbook

• Approximate Reading for Todays Material
• Pages 71-83, 102-112
• Approximate Reading for Next Class
• Pages 123-127, 132-145

3
2 Views of Normal Fitting

• Fit Model to Data
• Choose .
• Fit Data to Model
• First Standardize Data
• Then use Normal .
• Note same thing, just different rescalings
• (choose scale depending on need)

4
Normal Distribution Notation

• The normal distribution,
• with mean standard deviation
• is abbreviated as

5
Interpretation of Z-scores

• Recall Z-score Idea
• Transform data
• By subtracting mean dividing by s.d.
• To get (mean
  0, s.d. 1)
• Interpret as
• I.e. is sds above the mean

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Interpretation of Z-scores

• Same idea for Normal Curves
• Z-scores are on scale,
• so use areas to interpret them
• Important Areas
• Within 1 sd of mean
• the majority

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Interpretation of Z-scores

• Within 2 sd of mean
• really most
• Within 3 sd of mean
• almost all

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Interpretation of Z-scores

• Interactive Version (used for above pics)
• From Publishers Website
• http//bcs.whfreeman.com/ips5e/
• Statistical Applets
• Normal Curve

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Interpretation of Z-scores

• Summary
• These relations are called the
• 68 - 95 - 99.7 Rule
• HW 1.86 (a 234-298, b 234, 298),
• 1.87

10
Computation of Normal Areas

• Classical Approach Tables
• See inside covers of text
• Summarizes area computations
• Because cant use calculus
• Constructed by computers
• (a job description in the early 1900s!)

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Computation of Normal Areas

• EXCEL Computation
• works in terms of lower areas
• E.g. for
• Area lt 1.3
• is 0.7257

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Computation of Normal Areas

• Interactive Version (used for above pic)
• From Same Publishers Website
• http//bcs.whfreeman.com/ips5e/
• Statistical Applets
• Normal Curve

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Computation of Normal Areas

• EXCEL Computation
• (of above e.g.)
• Use NORMDIST
• Enter parameters
• x is cutoff point
• Return is Area below x

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Computation of Normal Areas

• Computation of areas over intervals
• (use subtraction)
• -

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Computation of Normal Areas

• Computation of areas over intervals
• (use subtraction for EXCEL too)
• E.g. Use Excel to check 68 - 95 - 99.7 Rule
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg9.xls

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Normal Area HW

• HW (use Excel)
• 1.94
• 1.97 (Hint the above 130
• 100 - below
  130)
• 1.99 (see discussion above)
• 1.113
• Caution Dont just twiddle EXCEL until answer
  appears. Understand it!!!

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And Now for Something Completely Different

• A mind blowing video clip
• 8 year old Skateboarding Twins
• http//www.youtube.com/watch?v8X2_zsnPkq8modere
  latedsearch
• Do they ever miss?
• You can explore farther
• Thanks to Devin Coley for the link

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Inverse of Area Function

• Inverse of Frequencies Quantiles
• Idea Given area, find cutoff x
• I.e. for
• Area 80
• This x
• is the quantile

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Inverse of Area Function

• EXCEL Computation of Quantiles
• Use NORMINV
• Continue Class Example
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg9.xls
• Probability is Area
• Enter mean and SD parameters

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Inverse Area Example

• When a machine works normally, it fills bottles
  with mean 25 oz, and SD 0.2 oz.
• The machine is out of control when it
  overfills. Choose an alarm level, which will
  give only 1 false alarms.
• Want cutoff, x, so that Area above 1
• Note Area below 100 - Area above 99
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg9.xls

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Inverse Area HW

• 1.95, 1.101, 1.107, 1.109
• 1.116 a (-0.674, 0.674)
• 1.117
• 1.118 (4.3)

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Normal Diagnostic

• When is the Normal Model good?
• Useful Graphical Device
• Q-Q plot Normal Quantile Plot
• Idea look at plot which is approximately linear
  for data from Normal Model

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Normal Quantile Plot

• Approach, for data
• Sort data
• Compute Theoretical Proportions
• Compute Theoretical Z-scores
• Plot Sorted Data (Y-axis) vs.
• Theoretical Z scores (X-axis)

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Normal Quantile Plot

• Several Examples
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg12.xls
• Show how to compute in Excel
• Steps as above

25
Normal Quantile Plot

• Main Lessons
• Melbourne Winter Temperature Data
• Gaussian is good, so looks linear
• So OK, to use normal model for these data
• Adding trendline helps in assessing linearity

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Normal Quantile Plot

• Main Lessons
• Intro Stat Course Exam Scores Data
• Skewed distributions ?? nonlinearity
• Outliers show up clearly
• Normal model unreliable here
• Combined plot highlights
• Mean Y-intercept
• Standard Deviation Slope

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Normal Quantile Plot

• Main Lessons
• Simulated Bimodal Data
• Curve is flat near modes
• Roughly linear near peaks
• Corresponds to two normal subpopulaitons
• Goes up fast a valley

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Normal Quantile Plot

• Homework
• 1.122
• 1.123
• 1.125

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And now for something completely different

• Recall
• Distribution
• of majors of
• students in
• this course

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And now for something completely different

• How about a biology joke?
• A seventh grade Biology teacher arranged a
  demonstration for his class. He took two earth
  worms and in front of the class he did the
  following He dropped the first worm into a
  beaker of water where it dropped to the bottom
  and wriggled about. He dropped the second worm
  into a beaker of Ethyl alchohol and it
  immediately shriveled up and died. He asked the
  class if anyone knew what this demonstration was
  intended to show them.

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And now for something completely different

• He asked the class if anyone knew what this
  demonstration was intended to show them.
• A boy in the second row immediately shot his arm
  up and, when called on said "You're showing us
  that if you drink alcohol, you won't have worms."

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Variable Relationships

• Chapter 2 in Text
• Idea Look beyond single quantities, to how
  quantities relate to each other.
• E.g. How do HW scores relate
• to Exam scores?
• Section 2.1 Useful graphical device
• Scatterplot

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Plotting Bivariate Data

• Toy Example
• (1,2)
• (3,1)
• (-1,0)
• (2,-1)

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Plotting Bivariate Data

• Sometimes
• Can see more
• insightful patterns
• by connecting points

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Plotting Bivariate Data

• Sometimes
• Useful to switch off
• points, and only
• look at lines/curves

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Plotting Bivariate Data

• Common Name Scatterplot
• A look under the hood
• EXCEL Chart Wizard (colored bar icon)
• Chart Type XY (scatter)
• Subtype conrols points only, or lines
• Later steps similar to above
• (can massage the pic!)

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Scatterplot E.g.

• Data from related Intro. Stat. Class
• (actual scores)
• How does HW score predict Final Exam?
• HW, Final Exam
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg10.xls
• In top half of HW scores
• Better HW ? Better Final
• For lower HW
• Final is much more random

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Scatterplots

• Common Terminology
• When thinking about X causes Y,
• Call X the Explanatory Var. or Indep. Var.
• Call Y the Response Var. or Dep. Var.
• (think of Y as function of X)
• (although not always sensible)

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Scatterplots

• Note Sometimes think about causation,
• Other times Explore Relationship
• HW 2.1

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Class Scores Scatterplots

• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg10.xls
• How does HW predict Midterm 1?
• HW, MT1
• Still better HW ? better Exam
• But for each HW, wider range of MT1 scores
• I.e. HW doesnt predict MT1 as well as Final
• Outliers in scatterplot may not be outliers in
  either individual variable
• e.g. HW 72, MT1 94
• (bad HW, but good MT1?, fluke???)

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Class Scores Scatterplots

• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg10.xls
• How does MT1 predict MT2?
• MT1, MT2
• Idea less causation, more exploration
• Still higher MT1 associated with higher MT2
• For each MT1, wider range of MT2
• i.e. not good predictor
• Interesting Outliers
• MT1 100, MT2 56 (oops!)
• MT1 23, MT2 74 (woke up!)

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Important Aspects of Relations

• Form of Relationship
• Direction of Relationship
• Strength of Relationship

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I. Form of Relationship

• Linear Data approximately follow a line
• Previous Class Scores Example
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg10.xls
• Final vs. High values of HW is best
• Nonlinear Data follows different pattern
• Nice Example Bralowers Fossil Data
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg11.xls

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Bralowers Fossil Data

• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg11.xls
• From T. Bralower, formerly of Geological Sci.
• Studies Global Climate, millions of years ago
• Ratios of Isotopes of Strontium
• Reflects Ice Ages, via Sea Level
• (50 meter difference!)
• As function of time
• Clearly nonlinear relationship

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II. Direction of Relationship

• Positive Association
• X bigger ? Y bigger
• Negative Association
• X bigger ? Y smaller
• E.g. X alcohol consumption, Y Driving
  Ability
• Clear negative association

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III. Strength of Relationship

• Idea How close are points to lying on a line?
• Revisit Class Scores Example
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg10.xls
• Final Exam is closely related to HW
• Midterm 1 less closely related to HW
• Midterm 2 even related to Midterm 1