Chapter Overview

1
Chapter Overview

• Statistical Process
• Know your question plan your study or
  experiment
• ID Population parameter of interest
• Plan your study (using historical data cant
  claim causation)
• Or plan your experiment (control, random
  selection, random assignment to control,
  replication)
• EAC
• Evidence (collect data)
• Analysis (graph calculate statistics)
• Conclusion (only infer to population if SRS)
• Analysis (A of EAC) Present and describe sets
  of data
• Graph
• Qualitative Data (_attribute, or categorical
  data used to show relationships/proportions)
• Circle
• bar graph
• Pareto Diagram (bar graph with cumulative line
  graph_)
• Quantitative Data (uses graph to show_dispersion)
• dot plot
• stem leaf
• frequency chart

• Conclusion (C of EAC) Interpret findings so
  that we know what the data is telling us about
  the sampled population

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Summary 2.1-2.2 Data Presentation

• Categorical (Qualitative)
• Circle graph
• Bar graph
• Bars spaced (x axis is categories not measure)
• Pareto
• Prioritized columns descending
• Plots cumulative
• Exceptions
• Some quantitative data summarized into categories

• Quantitative
• Dot plot
• Vertical or horizontal
• Stem and leaf
• State leaf unit
• Large group subdivide into 2 or 5 subgroups
• Histogram
• Bar graph of frequency distr.
• Bars touch (x axis is a scale)
• Frequency distribution
• Ungrouped
• Grouped (f vs. midpoint)
• Relative ( vs. class boundaries)
• Cumulative
• Cumulative relative (Ogive vs upper limit of
  class)

Interpret Patterns Overlapping distributions?
Separate into back to backs (p.47) Normal,
symmetrical mirror image Uniform box
like Bimodal two peaks Skewed left negatively
skewed, mean to left of median Skewed right
positively skewed, mean to right of median J
shaped no tail on side with highest frequency
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2.3 2.4 Measures Overview (no rounding until
final answer round to one more place than data)

• 2.3 Measures of Center
• Mean
• Of sample
• Of population
• Depth of median (n1)/2
• Median MD
• Mode
• Midrange MR (hi low)/2
• Measures for grouped data
• mean
• Weighted mean
• Variance standard deviation

• 2.4 Measures of Dispersion
• Range hi - low
• Deviation from mean
• Abs. value of deviation
• Mean abs. deviation
• Sample variance
• Sample standard deviation s
• square root of variance
• (used to est. pop.std.dev.-underestimates so n-l
  to reduce bias)
• Population standard deviation

SECS S hape unimodal, bimodal, symmetric,
skewed ( or -) E xceptions 1.5 IQR meaning
Q1-1.5(stand.dev.) or Q31.5(stand.dev.) C
enter (measures of central tendency) mean,
median, mode, midrange S pread (measures of
dispersion) range, interquartile range,
var., standard deviation RESISTANT
Measures mean or median? variance or standard
deviation?
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Remember Chapter 1

• Only experiments can conclude causation
• 3 Requirements for Exp.
• Control
• May have 2 groups being compared (comparative
  exp.)
• Direct manipulation of independent variable
• measure the effect on the dependent or response
  variable in a quantified way
• Randomization
• Statistical equivalence (subjects must be as
  similar as possible) RANDOM SELECTION
• RANDOM ASSIGNMENT to control or treatment group
• Control of extraneous variables
• Replication
• Repeat with large s so chance variation can be
  reduced and the effect of treatment more easily
  identified
• Process
• Purpose of research
• Plan define var., pop.,sample, methods
• Collect data
• Analyze data

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Additional Tips From Prep Workbooks

• Outliers mean 1.5IQR or Q1-1.5(Q3-Q1) and
  Q31.5(Q3-Q1)
• Dot plots ideal for discrete data
• can you find. from hist., cum.rel freq., etc.
  means is it possible even if it takes multi.
  steps
• Which is best? Hist., stem leaf means which
  one can be pulled from the quickest
• cut-point class mark for grouped histogram
• Data set could be 5,5,5,5 where box plot would
  be just a mark at 5
• When asked how change affects measures model
  with your own s if needed
• 
  
• 
  can use
  99.73std.dev.(which is 6 sigma) to verify
  reasonable std. dev.
• 
  
  
• 
  
  
  Standardized normal curve (meaning datas been
  translated to z scores) always
  
  looks exactly the same
  (ht spread) - center is always z0 stand.
  dev. 1 (always)
• 
  
• 
  Normal curve is denser at center (greater
  area under the curve for 1 increment)
• 
  When using actual
  data (not z s) small standard deviation shorter
  range so taller
• 
  
  large standard deviation
  wider range so flatter

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Warm Up after chapter 2 Name_______________

• When deciding if a variable is quantitiative or
  qualitative, the most important criteria to
  consider is _____________________________.
• When describing a distribution remember to
  discuss ______________________.
• Describing the ____________ can be very
  meaningful for a histogram but may not be for a
  bar graph of _____________ data.
• A study conducted on youth obesity collected data
  via a pedometer attached to each child.
  Researchers found some of the more obese children
  had the highest activity levels. How do you
  think this could be explained?
• In algebra we use x to represent ____________
  which has ____________ value(s). In stats we
  use X to represent ______________which has
  _____________value(s).
• In algebra x x __ ____________ in stats
  X X ____ _____________
• Game of Greed. Roll a pair of dice. All stand.
  You can take the sum or remain standing and add
  the sum of the next roll. However, when the sum
  of 2 is rolled the round ends, and you score 0.
  Play 5 rounds and collect data.

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Chapter 3 Descriptive Analysis Presentation
of Bivariate Data

• To be able to present bivariate data in tabular
  and graphic form

• To become familiar with the ideas of descriptive
  presentation

• To gain an understanding of the distinction
  between the basic purposes of correlation
  analysis and regression analysis

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Sept. 8, Tues.   P. 140 3.11,12,15,17,18,22 Wed.  
        Read p. 150, work p. 152 3.35,36, 37 and
39 Thurs.       Section 3.3 work p.165
56-62 Fri.           Quiz C3 hwk chapter
exercises 3.70, 3.76 and 3.86
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Vocabulary

• Bivariate data
• Sample statistics
• 3 data type combinations for bivariate data
• Contingency table
• Cross-tabulation
• Scatter-plot 1st step to see if linear
  relationship exists between 2 quantitative
  variables. predicted value is on y-axis
• Independent variable
• Dependent variable
• Ordered pair
• Input variable
• Output variable
• Least squares regression
• Line of best fit
• Linear correlation
• Linear regression
• Residuals plot
• Correlation analysis
• Positive correlation
• Negative correlation

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3.1 Bivariate Data

• Bivariate Data Consists of the values of two
  different response variables that are obtained
  from the same population of interest.

• Three combinations of variable types
• Both variables are qualitative (_________or_______
  ___) arranged on a
• a. ____________ or ____________ table (These
  statistics may also be displayed in a
  side-by-side bar graph)
• b. Example A survey was conducted
  to investigate the relationship between
  preferences for television, radio, or newspaper
  for national news, and gender. The results are
  given in the table below
• This table may be extended to
  display the _____totals (or _______). The total
  of the marginal totals is the grand total
• Note contingency tables often show percentages
  (relative frequencies). These percentages are
  based on the entire sample or on the subsample
  (row or column) classifications.
• One variable qualitative (_______) and other
  quantitative (________)
• Quantitative values viewed as separate samples
• each set identified by levels of the qualitative
  variable
• Statistics for comparison measures of central
  tendency, measures of variation, 5-number summary
• Each is described using techniques from C2
  results displayed side by side for easy
  comparison (dot-plots or box plots)
• Both variables are quantitative (both numerical)
• a. Expressed as _____________ (__,__)
• b. x _________variable, __________
  variable y _________variable, ___________
  variable
• c. Present data pictorially on a
  _________diagram

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Illustration

• These sample statistics (numerical values
  describing sample results) can be shown in a
  (side-by-side) bar graph

TV Radio NP
Percentages Based on Row (Column) Totals

• The entries in a contingency table may also be
  expressed as percentages of the row (column)
  totals by dividing each row (column) entry by
  that rows (columns) total and multiplying by
  100. The entries in the contingency table below
  are expressed as percentages of the column totals

Note These statistics may also be displayed in a
side-by-side bar graph
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Example

• Example A random sample of households from
  three different parts of the country was obtained
  and their electric bill for June was recorded.
  The data is given in the table below

• The part of the country is a qualitative variable
  with three levels of response. The electric bill
  is a quantitative variable. The electric bills
  may be compared with numerical and graphical
  techniques.

. . . . . . .
. ----------------------------------
----------------- Northeast
. ... ..
----------------------------------------
----------- Midwest
.
. . . . .
. . -----------------------------
---------------------- West
24.0 32.0 40.0 48.0 56.0
64.0

• The electric bills in the Northeast tend to be
  more spread out than those in the Midwest. The
  bills in the West tend to be higher than both
  those in the Northeast and Midwest.

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Two Quantitative Variables
Scatter Diagram The first tool used in
determining whether a _______ __________ exists
between 2 _______ _____________. Decide which
variable is to be _________. This variable will
be the __________ variable. Plot of all the
____________of _________ data on a coordinate
axis system. The input variable __ is plotted on
the __________ axis, and the output variable __
is plotted on the _________axis.
Note Use scales so that the range of the
y-values is equal to or slightly less than the
range of the x-values. This creates a window
that is approximately square.

• Example In a study involving childrens fear
  related to being hospitalized, the age and the
  score each child made on the Child Medical Fear
  Scale (CMFS) are given in the table below

Construct a scatter diagram for this data.
age input variable, CMFS output variable
How to construct a scatter diagram  1.
Find the range of x values range of y values.
2. Then choose your increments for x-axis
y-axis (theyre not always the
same) 3. Plots the points - ordered pairs
(x,y). 4. Label both axes give a title
to the diagram.
examples 1. the number of hours studied for an
exam versus the grade on the exam, 2. the number
of years a runner has been training versus
his/her time for running a mile, 3. the weight
of a car versus its gas mileage. 4. Age and
price of a car Determine whether the dependent
variable in each example will increase or
decrease as the independent variable increases.
What examples can you come up with on your own.?
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Take a look.

• AGAINST ALL ODDS Inside Statistics has three
  videos presenting the concepts of correlation and
  regression analysis for bivariate data. Program
  9 "Correlation" reinforces the concepts behind
  correlation with several excellent examples.
  Program 8 "Describing Relationships" and the
  first 15 minutes of Program 7 "Models for
  Growth" give additional insight into regression
  analysis plus excellent examples.
• The Student Suite CD contains three video clips
  Bivariate Data, Linear Correlation and
  Linear Regression.
• Paper helicopter link http//courses.ncssm.edu/ma
  th/Stat_inst01/intro.htm

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3.2 Linear Correlation

• Linear Correlation Measures the strength of a
  linear relationship between 2 variables

• As x increases, no definite shift in y no
  correlation

• As x increases, a definite shift in y correlation

• __________ correlation x increases, y increases

• ___________correlation x increases, y decreases

• If the ordered pairs follow a straight-line path
  __________ correlation
• If the points are patterned close to the line
  _________
• If the points are spread out, yet still look
  linear _____________

• Perfect positive correlation all the points lie
  along a line with positive slope

• Perfect negative correlation all the points lie
  along a line with negative slope

• If the points lie along a _________ or _________
  line no correlation

• If the points exhibit some other nonlinear
  pattern no linear relationship, no correlation

• Need some way to measure correlation

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Measures of Correlation
Coefficient of Linear Correlation r, measures
the strength of the linear relationship between
two variables

• Notes
• r 1 perfect positive correlation
• r -1 perfect negative correlation

Alternate Formula for r
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Example

• Example The table below presents the weight (in
  thousands of pounds) x and the gasoline mileage
  (miles per gallon) y for ten different
  automobiles. Find the linear correlation
  coefficient

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Please Note

• r is usually rounded to the nearest hundredth

• r close to 0 little or no linear correlation

• As the magnitude of r increases, towards -1 or
  1, there is an increasingly stronger linear
  correlation between the two variables

• Method of estimating r based on the scatter
  diagram. Window should be approximately square.
  Useful for checking calculations.
• Can you make any predictions about circumstances
  based on the various outputs? If age vs. price
  has r -0.9, then ___
• r only measures the strength of a linear
  relationship, and a cause and effect relationship
  cannot be concluded
• Before answering any questions concerning data in
  contingency tables, add all of the rows and
  columns. Be sure the sum of the row totals the
  sum of the column totals the grand total. Now
  you are ready to answer all questions easily.
• Explanatory variable independent x
• Response variable dependent y what your e
  predicting
•  

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3.2 Linear Correlation Understanding the
Linear Correlation Coefficient

• Estimating r - the linear correlation coefficient
•  
• 1. Draw as small a rectangle as possible that
  encompasses all of the data on the scatter
  diagram. (Diagram should cover a "square window"
  - same length and width)
• 2. Measure the width.
• 3. Let k the number of times the width fits
  along the length or in other words
  length/width.
• 4. r (1 - 1/k )
• 5. Use , if the rectangle is slanted positively
  or upward.
• Use -, if the rectangle is slanted negatively
  or downward.
• If there is a strong linear correlation between
  two variables, then one of the following
  situations may be true about the relationship
  between the two variables.
•  There is a direct cause-and-effect relationship
  between the two.
• There is a reverse cause-and-effect relationship.
• Their relationship may be caused by a third
  variable (called a ______ ________).
• Their relationship may be caused by the
  interactions of several other variables.
• The apparent relationship may be strictly a
  coincidence.
• Remember that a strong correlations does not
  necessarily imply causation.

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Text Problems

• P.140 3.11, 12, 15,17, 18, 22
• Read p. 150 lurking variables causation
• P.152 3.35, 36,37

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Key Homework Problem Solutionsfor Small Group
Discussion Correction

• Problem 1
• Problem 2
• Problem 3
• Problem 4
• Write your personal reflections about the
  statistics you learned from doing the problem to
  turn in for your warm up activity.

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3.3 Linear Regression

• If a linear relationship exists between two
  variables, that is,
• 1. its scatter diagram suggests a
  __________ _______________
• 2. its calculated ____ value is not near
  _________
•  
• Linear regression will calculate an __________
  of a ________based on the data. . This line,
  also known as the ____________ of ___________
  _________, will fit through the data with the
  smallest possible amount of ________ between it
  and the actual data points. The regression line
  can be used for generalizing and _________ over
  the sampled ________ of x.
• STASTICAL FORM OF A LINEAR REGRESSION LINE
• where _____________
  format differs
  from algebra
• _____________
• _____________
• x ________________
• Regression analysis finds linear equation that
  best describes relationship between 2 variables
  .
• Least squares criterion Find the constants b0
  and b1 such that the sum is
  as small as possible
• Some examples of various possible relationships

What would a scatter diagram look like to suggest
each relationship?
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Illustration

• Observed and predicted values of y

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Example

• Example A recent article measured the job
  satisfaction of subjects with a 14-question
  survey. The data below represents the job
  satisfaction scores, y, and the salaries, x, for
  a sample of similar individuals

1) Draw a scatter diagram for this data 2) Find
the equation of the line of best fit
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Line of Best Fit
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Scatter Diagram
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Sept. 29, Mon.  C3 work day p.174 practice test
Tues. AP practice problems Quiz Wed.  start
test Thurs.  finish C3 Test Fri.    activity
Design of Experiments/Studies Oct.6, Mon.  AP
practice problems
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Please Note

• Keep at least three extra decimal places while
  doing the calculations to ensure an accurate
  answer

• When rounding off the calculated values of b0 and
  b1, always keep at least two significant digits
  in the final answer

• The slope b1 represents the predicted change in y
  per unit increase in x

• The y-intercept is the value of y where the line
  of best fit intersects the y-axis

• One of the main purposes for obtaining a
  regression equation is for making predictions
• For a given value of x, we can predict a value of
  .
• The regression equation should be used to make
  predictions only about the population from which
  the sample was drawn
• The regression equation should be used only to
  cover the sample domain on the input variable.
  You can estimate values outside the domain
  interval, but use caution and use values close to
  the domain interval.
• Use current data. A sample taken in 1987 should
  not be used to make predictions in 1999.
• Work p. 165 3.57, 58, 59, 60 and 61

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Chapter Practice
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Supplemental Material
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List Serv Comments

• 1) What is plotted on a residuals plot? Is it the
  x-variable versus the residuals? If so, then in
  BVD book, on p. 149, they plot "predicted vs.
  gtresiduals." Is this another way to do
  it?Either way is fine. Typically in AP Stat it
  is x-variable v. residuals, though there was an
  AP Exam question once that did it the other way.
  Predicted v. residuals is needed when doing
  multiple regression.gt2) For TI-84 users, what
  is the difference between 4 LinReg(axb) versus
  8LinReg(abx), which is the one we use for
  stats? Both are found gtin the STAT CALC
  menu.They're the same, but abx is more
  commonly used for stats.gt3) Any good, short and
  sweet ways to explain regression to the
  mean?One thing you can do is take a scatterplot
  of something like 'Son's Height v. Father's
  Height' into several vertical bands. Put a big X
  at about the mean y-value for each vertical band.
  The regression line roughly passes through those
  bands, and the line will be less steep than the
  line yx.One way to explain the reason for this
  is to see that the line yx is approximately a
  symmetry line for the elliptical cloud of points.
  A segment perpendcular to yx with endpoints on
  the 'boundary' of the cloud is approximately
  centered on the line yx. But we are predicting
  vertically, not perpendicularly. Vertical
  segments with endpoints on the boundary of the
  cloud will be centered not on yx, but on a line
  less steep. That's hard to explain without a
  picture, but maybe it makes some sense? I
  hope?gt4) Why can we assume that the least
  squares regression line must go through the point
  (x-bar, y-bar)? Why can we assume that the mean
  gtvalue of the x-variable must necessarily
  correspond to the mean value of the
  y-variable?It's not that we assume this, it
  just happens to fall out of the algebra when you
  minimize the sum of squared residuals. David Bee
  could probably reproduce that algebra easily. It
  would take me a while. And I'd probably have to
  look it up anyway!

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• Lisa and OthersLisa asks3) Any good, short
  and sweet ways to explain regression to the
  mean?4) Why can we assume that the
  least-squares regression line must go through the
  point (x-bar, y-bar)? Why can we assume that the
  mean value of the x-variable must necessarily
  correspond to the mean value of the
  y-variable?Re L3 Here's a fairly short
  explanation that will probably seem       long
  because it's being written outWe know one way
  of writing the equation of the regression line
  is       (y - ybar)/sy r(x - xbar)/sx, or
  simply z_y rz_xConsider 0 lt r lt 1 and
  consider an x higher than its mean xbar.Thus,
  z_x gt 0, and so so is rz_x, which means the
  corresponding yvalue is higher than its mean
  ybar. But since r lt 1, it followsthat z_y lt z_x,
  and so the predicted y value is closer to ybar
  thanx is to xbar so, for an observation with a
  high x value, we predicty to be high, but,
  relative to the standard deviations, not as
  highas x, and so the predictions all appear to
  regress toward the mean.

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• Re L4 At the APStat level, one justification
  would be as follows       Prerequisite The
  point-slope equation of a line.    Consider an
  equation of the form y a bx, where a and
  b    are to be determined by the method of
  least-squares but    the calculations are all
  automated now. The process involves   
  least-squares leading to normal equations, but
  consider all    n x_i values and their
  corresponding y_i values.     Thus, we would
  have                         yi a
  bxi       Summing both sides of this equation
  gives                     SUM yi an b SUM
  xi    Dividing by n gives                      
  ybar a b xbar       (1)    Since the
  equation of the line is                        
  y  a bx           (2)    subtracting (1)
  from (2) gives                                  
     y - ybar b(x - xbar)    which shows the
  line passes through the point (x-bar,y-bar).   
  Note This is just a justification and not a
  proof I think    someone in the Forum in the
  past gave a good non-calculus-using    proof but
  I don't recall it.HTH-- David BeePS
  Note we didn't determine the value of b. However,
  if interested,    since we have SUM yi an b
  SUM xi, we could get the second    normal
  equation by multiplying yi a bxi through by
  xi and    summing, giving SUM xiyi a SUM xi
  b SUM (xi)2. Since we    now have two equations
  in a and b, they could be solved for a    and b,
  which CALC Choice 8 LinReg(abx) in effect does
  for us.

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• I am writing to see if there is a quicker way to
  find the standard deviation of the residuals. I
  know the formula, but the only way I currently
  see how to do it is to 1) list data 2) find
  linreg 3) place predicted data on list 4) find
  resid on next list 5) find square of resid list
  on next list 6) find sum of squared resid, then
  quickly divide by n-2 and take square root. Is
  there another way to do it on a TI 84?
• Thanks in advance for your help.

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Phone book Activity

• Chapter 3 Descriptive Analysis of Bivariate
  Data-phone books
•  
• Teams are used for this project. Each team is
  given a local phone book. Each team determines a
  question of interest for which the answer is a
  proportion and then determines a sampling method
  for answering it. Possible questions of interest
  include the proportion who list a name and no
  address, the proportion who use initials only,
  the proportion of last names that end in son,
  or the proportion of last names for which only
  one household has that last name. Different
  sampling methods are appropriate for different
  questions. Sample size calculation must be part
  of the design. Each team reports its results to
  the class. Often, the same question is asked by
  more than one team, so the issue of variability
  in results and the relationship to margin of
  error can be discussed.