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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

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

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

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

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

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

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

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