The Normal Distribution and Other Continuous Distributions

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Chapter 6

• The Normal Distribution and Other Continuous
  Distributions

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6.1 Continuous Probability Distributions

• Continuous Random Variables
• If X is a continuous RV, then P(Xa) 0, where
  a is any individual unique value
• Because X has ? individual unique values
• P(a ? X ? b) something nonzero where a to
  b represents an interval
• Normal is most important continuous probability
  distribution.

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6.2 Normal Distribution

• Also known as Gaussian Distribution
• Works close enough for a lot of continuous RVs.
• Works close enough for a few discrete RVs.
• Necessary for our inferential statistics.
• Bell-shaped and symmetric.
• All measures of central tendency are equal.
• In theory, X is continuous and unbounded.

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Normal RV

• Probabilities for discrete RV were given by a
  probability distribution function.
• Probabilities for continuous RV are given by a
  probability DENSITY function (pdf).
• Normal pdf requires you to know two parameters to
  find probabilities ? and ?.

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Finding Normal Probabilities

• Equation 6.1 fun but not useful.
• Like to have a table for each combination of ?
  and ?.
• Cant.
• Generate 1 table that can be used by everyone.
• Get everyone to convert or transform data so that
  it works with that one table!
• Transform X into Z

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6.3 Evaluating Normality

• The assumption of Normality is made all the time
  sometimes correctly so, and sometimes
  incorrectly so.
• Said another way not all continuous random
  variables are normally distributed.

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Checking Normality

• Text discusses two ways in this section (other
  ways discussed in Stat 2!)
• Compare what you know about the data to what
  you know about the normal distribution.
• Construct a normal probability plot.

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Comparing actual data to theory

• Central tendency actual data mean, median, and
  mode should be similar.
• Variability
• Is the interquartile range about equal to
  1.33the standard deviation?
• Is the range about equal to 6 times the standard
  deviation?

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Comparing actual data to theory

• Shape
• plot the data and check for symmetry.
• check to determine if the Empirical Rule applies.
• Sometimes samples are small--is the data
  non-normal or do you have a non-representative
  sample?

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Normal Probability Plot

• Best left to software.
• The straighter the line, the better the sample
  approximates a normal distribution.
• Systematic deviation from a straight line
  indicates non-normality.

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Plot Construction

• Order the data
• Use inverse normal scores transformation to find
  the standardized normal quantile for each data
  point.
• P(Z lt Oi) i/(n1)
• i.e. solve for Oi for the 1st data point and the
  second data point, etc.

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Plot Construction (cont.)

• Plot the data points
• actual values on the Y axis
• Standardized Normal Quantiles on the X axis
• A straight line demonstrates normality.
• A non-straight line demonstrates non-normality.