Chapter 3 Examining Relationships

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Chapter 3Examining Relationships

• Get the facts first, and then you can distort
  them as much as you please.
• Mark Twain

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3.1 Scatterplots

• Many statistical studies involve MORE THAN ONE
  variable.
• A SCATTERPLOT represents a graphical display that
  allows one to observe a possible relationship
  between two quantitative variables.

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Response Variable vs. Explanatory Variable

• Response Variable
• Measures an outcome of a study

• Explanatory variable
• Attempts to explain the observed outcomes

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Response Variable vs. Explanatory Variable

• When we think changes in a variable x explain, or
  even cause, changes in a second variable, y, we
  call x an explanatory variable and y a response
  variable.

• y
• Response
• Variable
• x
• Explanatory variable

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IMPORTANT!

• Even if it appears that y can be predicted from
  x, it does not follow that x causes y.
• ASSOCIATION DOES NOT IMPLY CAUSATION.

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When examining a scatterplot, look for an overall
PATTERN.

• Consider
• Direction
• Form
• Strength
• Positive association
• Negative association
• outliers

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Positive vs. Negative Association

• Positive Association
• (between two variables)
• Above-average values of one tend to accompany
  above-average values of the other
• Below-average values of one tend to accompany
  below-average values of the other

• Negative Association
• (between two variables)
• Above-average values of one tend to accompany
  below-average values of the other

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3.2 Correlation

• Describes the direction and strength of a
  straight-line relationship between two
  quantitative variables.
• Usually written as r.

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Facts About Correlation

• Positive r indicates positive association between
  the variables and negative r indicates negative
  association.
• The correlation r always fall between 1 an 1
  inclusive.
• The correlation between x and y does NOT change
  when we change the units of measurement of x, y,
  or both.
• Correlation ignores the distinction between
  explanatory and response variables.
• Correlation measures the strength of ONLY
  straight-line association between two variables.
• The correlation is STRONGLY affected by a few
  outlying observations.

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3.3 Least-Squares Regression

• If a scatterplot shows a linear relationship
  between two quantitative variables, least-squares
  regression is a method for finding a line that
  summarizes the relationship between the two
  variables, at least within the domain of the
  explanatory variable x.
• The least-squares regression line (LSRL) is a
  mathematical model for the data.

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Regression Line

• Straight line
• Describes how a response variable y changes as an
  explanatory variable x changes.
• Sometimes it is used to PREDICT the value of y
  for a given value of x.
• Makes the sum of the squares of the vertical
  distances of the data points from the line as
  small as possible.

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Residual

• A difference between an OBSERVED y and a
  PREDICTED y

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Some Important Facts About the LSRL

• It is a mathematical model for the data.
• It is the line that makes the sum of the squares
  of the residuals AS SMALL AS POSSIBLE.
• The point is on the line, where
  is the mean of the x values, and is the
  mean of the y values.
• The form is (N.B. b is the
  slope and a is the y-intercept.
• (On the regression line, a change of one standard
  deviation in x corresponds to a change of r
  standard deviations in y)

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Some Important Facts About the LSRL

• The slope b is the approximate change in y when x
  increases by 1.
• The y-intercept a is the predicted value of y
  when

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Coefficient of Determination

• Symbolism
• It is the fraction of the variation in the values
  of y that is explained by the least-squares
  regression of y on x.
• Measure of HOW SUCCESSFUL the regression is in
  explaining the response.

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Calculation of

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Example
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Example Solution
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Things to Note

• Sum of deviations from mean 0.
• Sum of residuals 0.
• r2 gt 0 does not mean r gt 0. If x and y are
  negatively associated, then r lt 0.

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Outlier

• A point that lies outside the overall pattern of
  the other points in a scatterplot.
• It can be an outlier in the x direction, in the y
  direction, or in both directions.

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Influential Point

• A point that, if removed, would considerably
  change the position of the regression line.
• Points that are outliers in the x direction are
  often influential.

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Words of Caution

• Do NOT CONFUSE the slope b of the LSRL with the
  correlation r.
• The relation between the two is given by the
  formula
• If you are working with normalized data, then b
  does equal r since
• When you normalize a data set, the normalized
  data has a
• mean 0 and standard deviation 0.

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More Words of Caution

• If you are working with normalized data, the
  regression line has the simple form
• Since the regression line contains the mean of x
  and the mean of y, and since normalized data has
  a mean of 0, the regression line for normalized x
  and y values contains (0, 0).