chapter 6: probability
DESCRIPTION
Chapter 6: Probability. Inferential Statistics. Flow of inferential statistics and probability. Sample. Population. Probability. Jars of Marbles. Population 1. Population 2. 50 black marbles 50 red marbles. 9 0 black marbles 1 0 red marbles. P(black) = .50. P(black) = .90. - PowerPoint PPT PresentationTRANSCRIPT
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Chapter 6: Probability
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Sample Population
Inferential Statistics
Probability
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Population 1 Population 2
50 black marbles50 red marbles
90 black marbles10 red marbles
P(black) = .50 P(black) = .90
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Population 1 Population 2
50 black marbles50 red marbles
90 black marbles10 red marbles
Sample of n = 4 selected. All 4 marbles are black. Which population did it come from?
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Probability of A
=Number of outcomes classified as A
Total number of possible outcomes
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Probability is Proportion
• Coin tosses -- p (heads) = ?• Cards• p (King of Hearts) = ?• p (ace) = ?• p (red ace) = ?
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6, 6, 7, 7, 7, 7, 8, 8, 8, 9
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x f9 18 37 46 2
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Random Sample
1. Each individual in the population has an equal chance of being selected.
2. If more than one individual is selected, there must be constant probability for each and every selection.
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Biased Sample
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What do we mean by a constant probability?
• Imagine selecting two cards from a deck
• First pick: P(Jack) = ?
• Second pic: P(Jack) = ? (It depends)
• Sampling with replacement (put the first card picked back in the deck)
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Probabilities for a range of scores
• In statistics we are often interested in computing probabilities for a range of scores from a distribution
• For example what is the probability of a score greater than 4?
• P(x > 4) = ?
• P(x < 3) = ?
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1, 1, 2, 3, 3, 4, 4, 4, 5, 6
x f
6 1
5 1
4 3
3 2
2 1
1 2
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What is the probability of a score greater than 4?
1 2 3 4 5 6 7 8
1
2
3
X
freq
uenc
y
P(x > 4) = ?
P(x > 4) = .20
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What is the probability of a score less than 3?
1 2 3 4 5 6 7 8
1
2
3
X
freq
uenc
y
P(x < 3) = ?
P(x < 3) = .30
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What is the probability of a score less than 3 or greater than 4?
1 2 3 4 5 6 7 8
1
2
3
X
freq
uenc
y
P(x < 3 or x > 4) = ?
P(x < 3 or x > 4) = .50
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So the proportion of area corresponding to a range of scores is the probability of
selecting a score within that range
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µ X
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13.59%
µ
z-2 -1 0 +1 +2
34.13%
13.59%
2.28%
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= 6
µX68 74 80
6.5 (a)
µ
z68 74 80
(b)
0 +2.00
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z-2 -1 0 +1 +2-3 +3
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(A)z
(B)Proportion in
Body
(C)Proportion in
Tail
0.00 0.5000 0.5000
0.01 0.5040 0.4960
0.02 0.5080 0.4920
0.03 0.5120 0.4880
0.21 0.5832 0.4168
0.22 0.5871 0.4129
0.23 0.5910 0.4090
0.24 0.5948 0.4052
0.25 0.5987 0.4013
0.26 0.6026 0.3974
0.27 0.6064 0.3936
0.28 0.6103 0.3897
0.29 0.6141 0.3859
Mean z
zMean
CC
B
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(c)
z0-.5µ
(a)
0 1.0z
µ
(b)
z0 1.5µ
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z
(a)
µ-0.40 0 1.25
µ
z
(b)
0.350 1.40
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X Z
Probability
z-score formula
Unit normal table
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= 100
X
µ
500 650
z0 1.50
If you select a score at random what is the probability of a score greater than 650?
?
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= 100
X
µ
500 700600
z
0 2.001.00
If you select a score at random what is the probability of a score between 600 and 700?
?
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= 100
Xµ = 500
z0
Top 15%
1.04
Find the 85th percentile score for this distribution
X = ?
?z scale
Lower 85%
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Xµ = 100
z0 1.40
= 10
114
If you select a score at random what is the probability of a score less than 114?
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Xµ = 100
z0-0.80
= 10
92
If you select a score at random what is the probability of a score less than 92?
?
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Xµ = 60
= 5
P = 0.3400or 34%
Find the 34th percentile score for this distribution
x = ?
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25% 25% 25% 25%z
Quartiles0 +0.67-0.67
Q1 Q2 Q3
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0.50
0.25
00 1 2
Pro
babi
lity
Number of heads in two coin tosses
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0 1 2 3 4
0.375
0.250
0.125
Pro
babi
lity
(a)
0 1 2 3 4 5 6
Pro
babi
lity
0.2500
0.1875
0.1250
0. 0625
(b)
Number of Heads in six coin tosses
Number of Heads in four coin tosses
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The Relationship Between the Binomial Distribution and the Normal Distribution
X0 1 2 3 4 5 6 7 8 9 10