Probability and Inferential Statistics

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Probability and Inferential Statistics Key
probability concepts Event Basic law of
probability Union of probabilities
(Or) Joint probabilities (And) Conditional
probabilities (Given) Independent events
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Example Problem 7.8 Crime Reported Not
reported Total Murder 12 12
24 Robbery 145 105 250 Assault
85 177 262 Rape 12 60 72 Auto
Theft 314 62 376 Total 568 416 984
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Example using conditional probabilities P(A and
B) P(A) P(BA) P(B) P(AB) Thus, P(BA)
P(B) P(AB)/P(A) Suppose are tested for a
disease and result is negative--you want to know
probability you are actually sick (false
negative) A negative result, P(A) 90
(.9) B sick, P(B) 1 in 10,000
(.0001) P(negativesick) P(AB) 5 (.05)
95 accurate Then P(BA) .0001 .05/.9
.0000056 Issues? Use other info?
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The Normal Probability Distribution The normal
or bell-shaped curve Symmetric and
unimodal Completely determined by mean (µ) and
standard deviation (s), note that mean
median mode A standard normal distribution is
a normal distribution with a mean of 0 and a s.d.
of 1 a variable with such a distribution is
denoted by Z If X is normally distributed with
mean µ and s.d. s, then Z (X - µ)/s has a
standard normal distribution
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The Normal Probability Distribution
(cont.) There is a known probability of values
occurring within a given number of standard
deviations of the mean 68.26 lie within 1
s.d. of mean 95.44 lie within 2 s.d. of
mean 99.72 lie within 3 s.d. of mean Thus,
using tables that contain the probability values
associated with the standard normal distribution,
it is possible to determine the probability (or
chance) that a particular occurrence of a
normally distributed variable will be close to
or far from the mean value This is one of the
two key foundations of inferential statistics
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An Example (p. 130 in MB) Z Table
Probability above Z 1.62 0.4474 5.26 0.73
2.40 -1.50 -0.4332 93.32 -0.48 -3.
16
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Another Example (problem 8.2 in MB) Data µ
27.2 minutes s 4.9 minutes What percentage
take between 22.3 and 32.1 minutes? Z1 (X1 -
µ)/s (22.3 - 27.2)/4.9 -1.00 Z2 (X2 -
µ)/s (32.1 - 27.2)/4.9 1.00 Table value
for Z 1.00 is .3413 Thus, percentage between is
(84.13 - 15.87) 68.26 What percentage take
between 17.4 and 37.0 minutes? What percentage
take between 12.5 and 41.9 minutes? (extra) What
percentage take between 20.34 and 34.06 minutes?