SLIDES PREPARED

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STATISTICS for the Utterly Confused, 2nd ed.

• SLIDES PREPARED
• By
• Lloyd R. Jaisingh Ph.D.
• Morehead State University
• Morehead KY

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14-2 The Chi-Square test for Goodness of Fit

• Hence one may intuitively conclude in this case
  that the sample data did not come from the
  population to which it is compared because of the
  large deviations from the sample distribution to
  the population distribution.

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14-4 Benfords Law

• Frank Benford, in the 1930s, noticed that
  logarithm tables (these were used by scientists
  long before the common use of computers and
  calculators) tended to be worn out on the early
  pages where the numbers started with the digit 1.
  

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14-4 Benfords Law

• Based on this observation and many others, he
  discovered that more numbers in the real world
  started with the digit 1 rather than with 2, and
  that more started with the digit 2 rather than
  with 3, and so on.
• He later published a formula which describes the
  proportion of times a number will begin with the
  digit 1, 2, 3, etc.

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14-4 Benfords Law

• This formula is now called Benfords Law.
• The Table on the next slide shows the
  distribution of the proportions, to three decimal
  places, for the leading digits of numbers based
  on Benfords Law.

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14-4 Benfords Law
The next slide shows a graphical Depiction of
Benfords Law.
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14-4 Benfords Law
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14-4 Benfords Law

• Example Students who attend college and apply
  for student loans must submit a FAFSA (Free
  Application for Federal Student Aid) form. Part
  of the information that is required is the annual
  income of the parent or parents. A sample of
  3,633 forms was sampled from a college records
  and the proportion, to three decimal places, of
  the leading digits for the total annual income
  for the parents were recorded. This information
  is presented on the next slide.

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14-4 Benfords Law

• Test at the 5 percent significance level whether
  the distribution of the first digits for the
  reported total salaries for the parents follow
  Benfords Law.

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14-4 Benfords Law

• Solution Plots of the proportions of the leading
  digits for both Benfords Law and the parents
  salaries are shown below.

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14-4 Benfords Law

• Solution (continued) The Table on the next slide
  shows the computations needed to compute the ?2
  test statistic.
• The value of the test statistic is equal to
  507.527.
• To obtain the expected frequencies based on
  Benfords Law one should multiply the total of
  3,633 by Benfords proportions.
• For example, from the table, the expected
  frequency value of 639.408 is obtained from
  3,6330.176 639.408, etc.

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14-4 Benfords Law
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14-4 Benfords Law
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EXAMPLE (Continued)

• Solution (continued) Diagram showing the
  rejection region.