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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

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?

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.

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.

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.

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 .

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

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

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

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

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

LESSON 4

- CONDITIONAL PROBABILITY

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?

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.

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 .

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?

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?

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

LESSON 5

- PERCENTILES OF ORDINAL VARIABLES

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.

STANDARD DEVIATION

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

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).

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.