PowerPoint Presentation Introduction to Probability and Statistics Eleventh Edition

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CS
PI
37
37
(a)
(b)
32
32
-115
-123
-115
-123
Before 1932-1999, M?3.0
Before 1984-1987, M?3.0
After 2000-2008, M?5.0
19/21 Hits
17/21 Hits
Is this difference statistically significant?
2
19/21
Cellular Seismology
17/21
Before 1984-1987, M?3.0
hits
Rundle et al. (PI)
Is this difference statistically significant?
Rates Changes in Rates
Before 1932-1999, M?3.0
After 2000-2008, M?5.0
area
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The Binomial Random Variable

• The coin-tossing experiment is a simple example
  of a binomial random variable. Toss a fair coin n
  3 times and record x number of heads.

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The Binomial Experiment

• The experiment consists of n identical trials.
• Each trial results in one of two outcomes,
  success (S) or failure (F).
• The probability of success on a single trial is p
  and remains constant from trial to trial. The
  probability of failure is q 1 p.
• The trials are independent.
• We are interested in x, the number of successes
  in n trials.

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9 probability
n 21 Coins
Fair Coin Null Hypothesis
P(observing 14 or more heads)
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4 probability
n 21 Coins
Fair Coin Null Hypothesis
P(observing 15 or more heads)
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21 probability
n 21 after earthquakes
PI Null Hypothesis (expect on average 17 hits)
P(observing 19 or more hits)
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1 probability
n 21 after earthquakes
P(observing 21 hits)
PI Null Hypothesis (expect on average 17 hits)
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Test of Statistical Significance

• Question Cellular Seismology (CS) - 19/21 hits,
  Pattern Informatics (PI) - 17/20 hits, but
• Does CS in general really perform better than PI,
  or are those extra two hits, just a coincidence?
• Need to decide between two possibilities
• The mean success rate of CS exceeds the mean
  success rate of PI.
• or
• ? The mean success rate of CS does not exceed the
  mean success rate of PI.
• This is an example of a statistical hypothesis
  test.

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Test of Statistical Significance

• Analogous to a courtroom trial. In trying a
  person for a crime, the jury needs to decide
  between one of two possibilities.
• The person is guilty.
• The person is innocent.
• Begin by assuming that the person is innocent.
• The prosecutor presents evidence, trying to
  convince the jury to reject the original
  assumption of innocence, and conclude that the
  person is guilty.

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Test of Statistical Significance

• The null hypothesis, H0
• Assumed to be true until we can prove
  otherwise.
• The alternative hypothesis, Ha
• Will be accepted as true if we can
  disprove H0.

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Reject the null hypothesis, H0, if (assuming that
H0 is true) the probability of observing the
number of hits that you actually did observe
(19/21) is low enough that it is unlikely to have
been observed due to only a random
occurrence. If 19/21 hits is unusually high, we
say that the difference between the expected
17/21 hits and what you actually observed is
statistically significant.
How low does the probability have to be? If
the probability is less than 0.05 (5), we say
that the difference is statistically significant
at the 95 level. If the probability is less
than 0.10 (10), we say that the difference is
statistically significant at the 90 level.
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n 21 after earthquakes
1 probability
PI Null Hypothesis (expect on average 17 hits)
P(observing 21 hits)
Statistically Significant or No way!
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n 21 after earthquakes
21 probability
PI Null Hypothesis (expect on average 17 hits)
P(observing 19 or more hits)
Not Statistically Significant or Way!
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