Download Chapter 18 Notes

Central Limit
Theorem
(Dr. Monticino)
Assignment Sheet


Read Chapter 18
Assignment #11 (Due April 13th)

Chapter 18
Exercise Set A: 1,2,3,5
Exercise Set B: 1,6
Exercise Set C: 2,3,5
Review Exercises: 1,2,4,5 , 7(which is right), 8,10,11

Test 2 on Wednesday April 13th over
Chapters 13 through 18
Overview

Central Limit Theorem

Intuition
Sums
Averages

Examples
Central Limit Theorem: Sums

For a large number of random draws, with
replacement, the distribution of the sum
approximately follows the normal
distribution

Mean of the normal distribution is
N* (expected value for one repetition)

SD for the sum (SE) is
N 

This holds even if the underlying population is
not normally distributed
Central Limit Theorem:
Averages

For a large number of random draws, with
replacement, the distribution of the average =
(sum)/N approximately follows the normal
distribution

The mean for this normal distribution is
(expected value for one repetition)

The SD for the average (SE) is

N

This holds even if the underlying population
is not normally distributed
Probability Histograms

In a probability histograms, the area of the bar
represents the chance of a value happening as a
result of the random (chance) process
 For many processes, the associated
probability histogram can be derived
explicitly
Draw two numbers from the box and sum or
average them.

For other processes, it may be difficult to
explicitly determine the probability
histogram
In many of these cases, the CLT indicates that the
normal distribution gives a good approximation
of the actual probability histogram

Draw 1000 numbers from the box and sum or
average them.
0
0
1
1
0
0
1
Empirical Histograms
Empirical histograms are histograms
based on actually performing an
experiment many times
 If the experiment is repeated many,
many times, then the empirical
histogram should be close to the
(theoretical) probability histogram

Example: Sums
Flip a fair
coin and
count the
number
of heads
– 10 flips
20%
Percent

15%
10%
5%
0%
2.00
4.00
6.00
Number of Heads
8.00
Example: Sums
Flip a fair
coin and
count the
number of
heads –50
flips
10.0%
7.5%
Percent

5.0%
2.5%
0.0%
15.00
20.00
25.00
30.00
Number of Heads
35.00
Example: Sums
Flip a fair
coin and
count the
number of
heads – 100
flips
8%
6%
Percent

4%
2%
40.00
50.00
Number of Heads
60.00
Example: Averages
Flip a fair
coin and
determine
the
proportion of
times heads
occurs – 10
flips
25%
20%
Percent

15%
10%
5%
0.00
0.20
0.40
0.60
Proportion of Heads
0.80
Example: Averages
Flip a fair
coin and
determine
the
proportion
of times
heads
occurs – 50
flips
10.0%
Percent

7.5%
5.0%
2.5%
0.0%
0.30
0.40
0.50
0.60
Proportion of Heads
0.70
Example: Averages
Flip a fair coin
and
determine the
proportion of
times heads
occurs – 100
flips
8%
6%
Percent

4%
2%
(Dr. Monticino)
0.30
0.40
0.50
0.60
Proportion of Heads