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LESSON 36 MATERIAL

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Title: LESSON 36 MATERIAL


1
LESSON 36 MATERIAL
  • Blackwell Library Circulation Desk
  • Luft
  • Lesson 36 Folder (Xerox 2)
  • Xerox Two Articles (5 sheets of paper)
  • Bring to class next time.
  • Copy second page under Journal Article Link
    (Revised May of 2003)

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LESSON 3
  • PROBABILITY

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1.5.2 Probability   EXAMPLE C3 A college of
1000 students has 700 day students (and the rest
night students). Also 600 of the students are
men (and the rest women). Also exactly 400
students are both men and day students. If we
choose one student at random, what is the
probability that the student chosen will be a
woman and a night student? What is the
probability that the student chosen will be a man
or a day student?
10
SOLUTION C3 In Example C1 we developed the
following table of counts
The probability of a randomly chosen student
being a woman and a night student is 100/1000
0.100 . The probability that the student
chosen will be a man or a day student is (300
400 200)/1000 0.900.
11
An alternative approach is to divide all the
counts in the table above by the table total
1000, giving a table of fractions or
probabilities.
Now the table of fractions tells us that a
randomly chosen student has probability 0.100
of being a woman and a night student. And
adding the numbers in the shaded region tells us
that a randomly chosen student has probability
0.300 .400 0.200 0.900 of being a man
or day student.
12
EXAMPLE D1 Often we are given fractions or
probabilities without counts, and we can draw
only the table of probabilities. Suppose 30 of
the freshman class smoke, 25 drink, and 10 both
smoke and drink. What is the probability that a
freshman selected at random is either a smoker or
a drinker (or both)? In answering such a
question, don't try to work it in your head.
SOLUTION D1 First find the events mentioned in
the problem, and then draw a table of
probabilities, without any of the given numbers.
13
Having drawn the table in a CORRECT way, we can
insert the fractions or probabilities in their
places, and complete the table by subtraction
the probability that a randomly chosen student is
a smoker or a drinker   P(S?D) 0.15
0.10 0.20 0.45   The probability that a
freshman chosen at random had exactly one of
these health-threatening behaviors (meaning not
both) is   0.15 0.20 0.35 .
14
Suppose we are given two events A and B with
probabilities P(A) 0.4 and P(B) 0.3 . This
is not enough information to compute the
probability P(A?B) of the union of A and B. We
must have additional information about the
probability of the intersection P(A?B). The table
below shows what we know.
Clearly we need other information to fill in all
the blanks.
M155 L35 Review of Probability -- Slide 1
15
If we are simply given the probability of the
intersection of A and B, as a number like
P(A?B) 0.06 We enter this probability 0.06 in
the table and compute the other entries by
subtraction to get
It follows that P(A?B) 0.24 0.06 0.34
0.64
M155 L35 Review of Probability -- Slide 1
16
If instead we know that A and B are mutually
exclusive they cannot occur at the same time,
then we have the probability of the
intersection P(A?B) 0 We enter this
probability 0 in the table and compute the
other entries by subtraction to get
It follows that P(A?B) 0.3 0 0.4 0.7
M155 L35 Review of Probability -- Slide 1
17
Finally, if we are told that the events A and B
are independent, we may use a simple formula to
compute the probability of the intersection
P(A?B) P(A) P(B) (0.4) (0.3) 0.12 We
enter this probability 0.12 in the table and
compute the other entries by subtraction to get
It follows that P(A?B) 0.18 0.12 0.28
0.58
M155 L35 Review of Probability -- Slide 1
18
EXAMPLE D2 Suppose 30 of the freshman class
smoke and 25 drink. If smoking is independent
of drinking in the freshman class, what is the
probability that a freshman selected at random is
either a smoker or a drinker (or both)?
  SOLUTION D2 Table total is 1, so internal
entries are products of marginal entries.
We add the fractions (probabilities) in the
shaded area to find the probability that a
randomly chosen student is a smoker or a
drinker   P(S?D) 0.175 0.075
0.225 0.475
19
LESSON 4
  • CONDITIONAL PROBABILITY

20
1.6.2 Conditional Probability     EXAMPLE C4 A
college of 1000 students has 700 day students
(and the rest night students). Also 600 of the
students are men (and the rest women). Also
exactly 400 students are both men and day
students. If we choose one student at random
from the women, what is the probability that the
student chosen will be a day student? a night
student?
21
SOLUTION C4 Since the restriction is to women,
we will show the women (and men) as a row in the
two-way table. We repeat the table from Example
C3 and show the computation of row fractions.
22
Of course, the ratios of integers in the table
above are merely indications of the row
fractions. What we really want is the decimal
fractions, like this
The shaded row is the set of women, to which the
choice is restricted in this problem. The
probability that a randomly chosen student will
be a day student is the row fraction 300/400
0.750 . Similarly, The probability that a
randomly chosen student will be a night student
is the row fraction 100/400 0.150 .
23
McClave Chapter 3 Exercise 3.41
(c) Given that both the attacker and the victim
were white, what is the probability that a
randomly selected reported crime involved
fatalities?
24
McClave Chapter 3 Exercise 3.41
(c) Given that both the attacker and the victim
were white, what is the probability that a
randomly selected reported crime involved
fatalities?
25
McClave Chapter 3 Exercise 3.41
(c) Given that both the attacker and the victim
were white, what is the probability that a
randomly selected reported crime involved
fatalities? 183/65210.0281
26
LESSON 5
  • PERCENTILES OF ORDINAL VARIABLES

27
LESSON 6
  • STANDARD DEVIATION SYMMETRY SKEWNESS

3.2 Dispersion   The dispersion of a variable is
the extent to which its scores are spread out.
The simplest measure of dispersion is the range
maximum - minimum the difference between the
highest and lowest scores. Also useful is the
interquartile range Q3-Q1 the difference
between the 75th percentile score and the 25th
percentile score. These two measures of
dispersion are insensitive to scores which are
not extreme.
28
STANDARD DEVIATION
29
3.3 Symmetry and Skewness   A histogram is
called symmetric about a level L when it can be
folded at L so the two parts of the scale of
levels come together and the two sides of the
histogram match. If a histogram is symmetric, it
is symmetric about a level which is
simultaneously the median and the mean the
median because half the subjects are below it and
half above it, and the mean because scores in the
histogram occur in paired strips equidistant from
the center L of symmetry.   A histogram which is
not symmetric is called skewed. The simplest
measure of skewness is proportional to the
difference between the mean and the median, and
we call it simple skewness.
But the mean cannot be farther from the median
than the amount of the standard deviation so that
30
BEST STATISTICAL METHOD
Both two-sample Wilcoxon tests require that the
population histograms differ by location only.
All t-tests require that either all populations
be normal, or all samples be large (or both).
31
FINAL EXAM
  • One or two number 2 pencils.
  • An eraser that won't smear
  • Calculator
  • Clean copy of journal article "Comparing Scores"
  • I will provide tables
  • Don't bring anything else.
  • Any cell phones must be turned off and put away.
  • Location of exam is on the web.
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