#### Transcript Slides 1-17 Normal and CLT

```BA 275
Agenda
 Quiz #1
 Experiencing Random Behavior


Normal Probability Distribution
Normal Probability Table
1
Review Question: Warranty Level
Mean = 30,000 miles STD = 5,000 miles
Q1: If the level of warranty is set at 15,000 miles, about what % of tires will be returned
under the warranty?
Q2: If we can accept that up to 2.5% of tires can be returned under warranty, what should be
the warranty level?
0.04
0.03
0.02
0.01
0
0
10
20
30
40
50
60
2
The Empirical Rule is not Enough
Mean = 30,000 miles STD = 5,000 miles
Q1: If the level of warranty is set at 12,000 miles, about what % of tires will be returned
under the warranty?
Q2: If we can accept that up to 3% of tires can be returned under warranty, what should be
the warranty level?
0.04
0.03
0.02
0.01
0
0
10
20
30
40
50
60
3
The Normal Probability Distribution
 A specific curve that is symmetric and bell-shaped with two
parameters m and s2.
 It has been used to describe variables that are too cumbersome
to be consider as discrete (i.e., continuous variable). For
example,
 Physical measurements of members of a biological
population (e.g., heights and weights), IQ and exam scores,
amounts of rainfall, scientific measurements, etc.
 It can be used to describe the outcome of a binomial experiment
when the number of trials is large.
 It is the foundation of classical statistics.
 Central Limit Theorem
4
Standard Normal Probabilities (Table
A)
5
Standard Normal Probabilities (Table
A)
6
Example 1
m =0
s=1
a = 1.96
A
Prob = ???
a
7
Example 2
C
m=0
s=1
a = ?????
Prob = 0.0793
a
8
Example 3
D
m=0
s=2
a = 2.00
b = ??????
Prob = 0.1005
a
b
9
Sampling Distribution (Section 4.4)
 A sampling distribution describes the
distribution of all possible values of a statistic
over all possible random samples of a
specific size that can be taken from a
population.
 45 
   3,169,000,000,000
 25 
10
Central Limit Theorem (CLT)
 The CLT applied to Means
If X ~ N ( m , s 2 ) , then X ~ N ( m ,
s2
).
n
If X ~ any distribution with a mean m, and variance s2,
then X ~ N ( m ,
CLT demo
s2
n
) given that n is large.
With a sample of size n = 25, can we predict the value of
the sample mean?
Example 1: X ~ a normal distribution with the mean 16, and variance 25.
Example 2: X ~ a distribution with the mean 8.08, and variance 38.6884.
11
Level
Mean = 30,000 miles STD = 5,000 miles
Q1: If the level of warranty is set at 15,000 miles, about what % of tires will be returned
under the warranty? => 0.15%
Q2: If we can accept that up to 2.5% of tires can be returned under warranty, what should be
the warranty level? => 20,000 miles
0.04
0.03
0.02
0.01
0
0
10
20
30
40
50
60
12
Answer: The Empirical Rule is not
Enough
Mean = 30,000 miles STD = 5,000 miles
Q1: If the level of warranty is set at 12,000 miles, about what % of tires will be returned
under the warranty? => almost 0.0000
Q2: If we can accept that up to 3% of tires can be returned under warranty, what should be
the warranty level? => 20,600 miles
0.04
0.03
0.02
0.01
0
0
10
20
30
40
50
60
13
m =0
s=1
a = 1.96
A
Prob = ???
a
Prob = 0.025
14
C
m=0
s=1
a = ?????
Prob = 0.0793
a
a = -1.41
15