Transcript Document

Probability
Introduction to Statistics
Chapter 6
Feb 5-10, 2009
Classes #8-9
Inferential Statistics



By knowing the make-up
of a population, we can
determine the
probability of obtaining
specific samples
In this way, probability
gives us a connection
between population and
samples
We try to use a sample
to make an inference
about the population

See marbles example
on pp. 135-136
Probability Definition

In a situation where several different
outcomes are possible, we define the
probability for any particular outcome
as a fraction or proportion
Probability Formula

P(event) =
# of outcomes classified as the event
Total # of possible outcomes
The probability of event A, p(A), is the
ratio of the number of outcomes that
include event A to the total number of
possible outcomes
Probability Definition

Example:
 What is the probability of obtaining an
Ace out of a deck of cards?
 p(Ace) = 4/52 = .07 or 7%
Probability Formula
P(event) =
# of outcomes classified as the event
total # of possible outcomes
Example:
What is the probability that a selected person has
a birthday in October (assume 365 days in a
year)?
Step 1: How many chances are there to have a
birthday in a year?
Step 2: How many chances are there to have a
birthday in October?
Step 3: The probability that a randomly selected
person has a birthday in October is:
P (October birthday) = 31/365 = 0.0849
Probability Definition
Probability values are contained in a
limited range (0-1).

If P=0, the event will never occur.

If P=1, the event will always occur.
1
Probability values are
contained in a limited
range (0-1).
• If P=0, the event will
never occur.
Certain
Likely
0.5
50-50
chance
Unlikely
• If P=1, the event will
always occur.
0
Impossible
Probability Definition
Probability can be expressed as fraction,
decimals or percentage.



P=3/4
P=0.75
P= 75%
*All of the above values are equal*
Simple (or Independent) Random
Sampling

Each individual in the population has an
equal chance of being selected


p = 1/N
If more than one individual is to be
selected for the sample, there must be
constant probability for each and
every selection
 In cards, 1/52 for each draw
Example 1:

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Selection and n =3 cards from full deck
of cards…
 p(Ace) = 4/52 (draw is a 3)
 p(Ace) = 4/51 (draw is a 2)
 p(Ace) = 4/50 (draw is a ACE)
If we drew a 4th card what would be the
probability?
 p(Ace) = ?/??
What is the problem with Example 1?

Remember our rules for simple random
sampling…
 How can we fix it?
 See next slide…
Sampling with replacement

To keep the probabilities from changing
from one selection to the next, it is
necessary to replace each sample
before you make the next selection
Example of Sampling with Replacement
You draw a name out of the hat and
record it, you put the name back
and it can be chosen again
Example 1 but this time with Sampling
with replacement :


Selection and n =3 cards from full deck
of cards…
 p(Ace) = 4/52 (draw is a 3)
 p(Ace) = 4/52 (draw is a 2)
 p(Ace) = 4/52 (draw is a ACE)
If we drew a 4th card what would be the
probability?
 p(Ace) = ?/??
Probability and Frequency
Distributions


Because probability and proportion are
equivalent, a particular proportion of a
graph will then correspond to a particular
probability in the population
Thus, whenever a population is
presented in a frequency distribution
graph, it will be possible to represent
probabilities as proportions of the
graph.
Example 2:

What is the
probability of
scoring 70 or
above on this
exam?

p(x > 70) = ?
Example 3
30
F
r
e
q
u
e
n
c
y
28
What is the probability of
drawing an exam of B or
better out of the pile of all 47
exams?
25
20
15
9
10
4
5
2
2
2
0
0
A
B+
B
C+
GRADE
C
D+
0
D
F
Probability and the Normal
Distribution
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
Statisticians often identify sections of a
normal distribution by using z-scores
Recall that z-scores measure positions
in a distribution in terms of standard
deviations from the mean


z = 1.00 is 1 SD above the mean
z = -2.00 is 2 SD’s below the mean
Probability and the Normal
Distribution

From the normal distribution graph you
can then determine the percentage of
scores falling above or below the zscore, between the mean and the zscore, or between 2 z-scores

See next slide
The Normal Curve
Mean = 65
S=4
0.70
99.72% of cases
Relative Frequency
0.60
95.44% of cases
0.50
68.26% of cases
0.40
0.30
0.20
0.10
0.00
2%
14%
34%
34%
14%
2%
51 53 55 57 59 61 63 65 67 69 71 73 75 77 79
-3S
-2 S
-1 S
0
+1 S
+2S
+3S
The normal distribution following a z-score
transformation
Note:
1. Left and right sides
of distribution have the
same proportions.
2. Proportions apply
to any normal
distribution.
Probability and the normal
distribution
• Why is this important?
– We can now
describe X-values
(raw scores) in
terms of probability.
• Example:
– What is the
probability of
randomly selecting
a person who is
taller than 80
inches?
SD = 6
68
74
80
What is the probability
of randomly selecting a
person who is shorter
than 74 inches?
The Unit Normal Curve Table
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Normal curve table gives the precise percentage of
scores between the mean (z score of 0) and any other z
score.
Can be used to determine:
 Proportion of scores above or below a particular z
score
 Proportion of scores between the mean and a
particular z score
 Proportion of scores between two z scores
NOTE: Using a z score table assumes that we are
dealing with a normal distribution
 If scores are drawn from a non-normal distribution
(e.g., a rectangular distribution) converting these to
z scores does not produce a normal distribution
The Unit Normal Table

The unit normal table lists relationships
between z-scores locations and
proportions in a normal distribution.

Table B.1 in Appendix B, pp. 527-530
Using Appendix B.1

Appendix B.1 has four columns (A), (B), and
(C) and (D)



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(A) = Z scores.
(B) = Proportion in body
(C) = Proportion in tail
(D) = Proportion between mean and z
The Unit Normal Table
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Column B: Proportion in body
The Unit Normal Table

Column C: Proportion in tail
Check this in Table B.1
Using Appendix B.1 to Find Areas
Below a Score
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Appendix B.1 can be used to find the areas
above and below a score.
First compute the Z score
Make a rough sketch of the normal curve and
shade in the area in which you are interested.
Check this in Table B.1


The area below Z = 1.67 is
0.4525 + 0.5000 = 0.9525.
Areas can be expressed as
percentages:
 0.9525 = 95.25%
The Unit Normal Table
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The body (column b) always
corresponds to the larger part of the
distribution whether it is on the righthand side or the left-hand side.
The tail (column c) is always the
smaller section whether it is on the
right or the left.
The Unit Normal Table
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The proportions on the right hand side
equals to the proportions on the left
hand side.
Proportions are always positive.
The two proportions (body and tail) will
always add to 1.00.
The Unit Normal Table
The table lists proportions of the normal
distribution for a full range of possible
z-score values.
See chart on the next slide
A portion of the unit normal table
The Unit Normal Table
Its important to know how to use it:
If you know the specific location (zscore) in a normal distribution, you
can use the table to look up the
corresponding proportions.
*Z-Score ~~~> Proportion*
And also….
The Unit Normal Table

If you know a specific proportion, you
can use the table to look up the exact
z-score location in the distribution.
*Proportion ~~~> Z-Score*
Probabilities, proportions, and
scores (X values)
The process…
1. Transform the x values into z-scores
(Chapter 5).
2. Use the unit normal table to look up the
proportions corresponding to the z-score
values.
See chart on the next slide
A map for probability problems
Example 4 – Road Race Results
Stacy ran:
Mean:
SD:
37 minutes
36 minutes
4 minutes
What percentage of people took longer
than Stacy to finish the race?
P(X>37) = ?
Example 5: What is the probability of randomly
selecting someone with an IQ score less than 120?
IQ scores
 Form a normal distribution
 With mean,  = 100
 SD,  = 15
Our question:
p(x<120)?
Example 5
Finding scores corresponding to
specific proportions or probabilities.

We can also find an x value that
corresponds to a specific proportion
in the distribution
 See next slide
Example 6
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What is the IQ score (X-Value) needed to be
in the top 80% of the distribution?
*This formula is now needed:
x= +z()
Example 7
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After an exam, you learn that the mean
for the class is 60, with a standard
deviation of 10. Suppose your exam
score is 70. What is your Z-score?
Where, relative to the mean, does your
score lie?
What is the probability associated with
your score (use Z table Appendix B.1)?
What if your score is 72?
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Calculate your Z-score.
What percentage of students have a
score below your score? Above?
What percentile are you at?
What if your mark is 55%?
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Calculate your Z-score.
What percentage of students have a
score below your score? Above?
What percentile are you at?
Another Question…
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
What if you want to know how much
better or worse you did than someone
else? Suppose you have 72% and
your classmate has 55%?
How much better is your score?
Probability:
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

Let’s say your classmate won’t show
you the mark….
How can you make an informed guess
about what your neighbor's mark might
be?
What is the probability that your
classmate has a mark between 60%
(the mean) and 70% (1 s.d. above the
mean)?
Credits
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http://publish.uwo.ca/~pakvis/The%20Normal%20Curve.ppt#289,
30,Probability:
http://homepages.wmich.edu/~malavosi/Chapter6PPT_S_05.ppt#
10