Lecture 3 Normal distribution, stem-leaf, histogram - PowerPoint PPT Presentation

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Lecture 3 Normal distribution, stem-leaf, histogram

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... histogram. Idealized Population, Box of infinitely many tickets, each ticket has a value. Random variable and probability statement P(X 85) ... – PowerPoint PPT presentation

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Title: Lecture 3 Normal distribution, stem-leaf, histogram


1
Lecture 3 Normal distribution, stem-leaf,
histogram
  • Idealized Population, Box of infinitely many
    tickets, each ticket has a value.
  • Random variable and probability statement P(Xlt85)
  • Notations , Greek letters Mean (expected value)
    and standard deviation, E(X) m , SD(X) s,
    Var(X) s2
  • Examples
  • Empirical distribution Stem-leaf, histogram
  • Three variants of histogram frequency, relative
    frequency, density(called standardized in book)
  • Same shape with different vertical scale
  • Density relative frequency / length of interval

2
  • Given a box of tickets with values that come from
    a normal distribution with mean 75 and standard
    deviation 15, what is the probability that a
    randomly selected ticket will have a value less
    than 85?
  • Let X be the number elected ( a random variable).
  • Pr( Xlt85).

3
How does the normal table work?
  • Start from Z0.0 , then Z0.1
  • Increasing pattern observed
  • On the negative side of Z
  • Use symmetry

4
How to standardize?
  • Find the mean
  • Find the standard deviation
  • Z (X-mean)/SD
  • Reverse questions
  • How to recover X from Z?
  • How to recover X from percentile?

5
  • Suppose there are 20 percent students failing the
    exam
  • What is the passing grade?
  • Go from percentage to Z, using normal table
  • Convert Z into X, using Xmean Z times SD

6
Probability for an interval
  • P (60ltXlt85)
  • Draw the curve (locate mean, and endpoints of
    interval)
  • P(Xlt85)-P(Xlt60) where
  • P(Xlt60) P(Zlt(60-75)/15)P(Zlt-1)1-P(Zlt1)1-.841
    about .16
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