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1
Lecture 6: Normal Distribution and
Statistical Inference
Psych 100A
Winter 2009
Normal distribution
1. General properties
2
Standard normal distribution
2. Standard normal distribution
3. Standardization of units
4. Finding areas under normal curves
Statistical Inference
1. Statistical inference: techniques
2. Point estimates
1
Example
• Is this a binomial experiment?
Yes
• Fair coin tossed four times. What is the chance of
getting exactly 2 heads?
1. n = 4 iid trials (assume coin toss process OK)
2. Dichotomous outcome (“heads, H” or “tails, T”)
3. Pr(H) = Pr(T) = ½, which is constant.
4. Variable = total number of successes (heads)
2
1
st
2
nd
Toss:
3
rd
4
th
HHTT
Solution 1.
•Draw
a
path
diagram.
HTHT
HTTH
1
2
7
6
3
4
5
8
H
T
H
T
H
T
H
T
H
T
H
T
T
H
H
3
6
THHT
THTH
TTHH
9
10
15
14
11
12
13
16
H
T
H
T
H
T
H
T
H
T
T
H
H
T
T
8
16
)
'
2
Pr(
s
H
3
Formula: Pr(success) in a binomial experiment
• Probability of x successes in n trials of a binomial
experiment is
x
n
x
p
p
x
n
x
n
successes
x
)
1
(
)!
(
!
!
)
Pr(
•where
• n = number of trials
• p = probability of success
• 1p = probability of failure
• x = number of successes
• x! = x(x1)(x2)…(2)(1)
4
4. Normal distribution
• Special continuous probability
distribution
5
Suppose
0
.125
.250
.375
Density
01 23
Number of heads
x
0
1
2
3
Pr(x)
.125
.375
.375
.125
N=3
N=10
x
0
1
2
3
4
5
Pr(x)
.0010
.0097
.0439
.1172
.2051
.2461
0
.125
.250
.375
01 2345678910
Number of heads
6
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Suppose
0
.025
.050
.075
.100
.125
Density
0 1
02
03
04
05
0
Number of heads
x
0
1
…
12
…
15
…
20
…
25
Pr(x)
.0000
.0000
…
.0001
…
.0020
…
.0418
…
1123
N=50
.1123
Number of heads
N=immense
An ideal or normal curve
should approximate well the
binomial distribution
7
.50
Normal curve
•
deMoivre’s normal curve
2
2
2
1
)
(
x
e
f
0
.25

4
3

2

101234
Standard units
8
1. General properties of normal dist.
a. Family of continuous probability distributions.
A particular member is defined by its center
(mean,
) and spread (SD,
).
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 Winter '10
 FIRSTENBERG,I.

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