Chapter 18
Context
Chapter 18
Probability
Histograms
Probability
Histograms
Central Limit
Theorem
Central Limit
Theorem
Histogram of sum of the draws, when repeated 1000 times
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Density
Chapter 18
The Normal Approximation for Probability Histograms
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Part V
Chance Variability
Recall the computer simulation of last lecture
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sum of the draws
Context
Chapter 18
Probability
Histograms
Probability histograms
Chapter 18
Probability
Histograms
Central Limit
Theorem
Central Limit
Theorem
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Density
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Histogram of sum of the draws, when repeated 1000 times
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60
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80
90
100
110
sum of the draws
If the number of draws from a box is relatively large, then the
sum and average of the draws follow a normal distribution.
This is useful, because we can then approximate probabilities by
using the normal curve.
We will now look at this phenomenon a bit closer.
The histograms that we saw so far are empirical histograms:
• They are based on data
• Area under the histogram represents the percentage of
cases
We'll now look at a new type of histogram: probability
histograms:
• They are not based on data, but on theory
• Area under the histogram represents chance
Denition
A probability histogram represents chance by area.
The total area under the histogram is 100%.
Empirical histograms converge to a
probability histogram
Chapter 18
Probability
Histograms
Central Limit
Theorem
If the number of repetitions is large (several thousands), then
the empirical histogram will look more and more like the
probability histogram
Example:
Central Limit
Theorem
Probability
Histograms
Central Limit
Theorem
Central limit theorem
Theorem (The Central Limit Theorem)
When drawing at random with replacement from a box, the
Empirical histograms converge to a
probability histogram
If the number of repetitions is large (several thousands), then
the empirical histogram will look more and more like the
probability histogram
We say that the empirical histogram converges to the
probability histogram.
Rolling two dice and looking at the sum.
Chapter 18
Probability
Histograms
Chapter 18
Example 1
Chapter 18
Probability
Histograms
Central Limit
Theorem
probability histogram for the sum will follow the normal curve,
even if the contents of the box do not. The histogram must be
put in standard units, and the number of draws must be
Sum of draws from the box
reasonably large.
The theorem applies to sums, but not to other operations
like products.
Note:
There is no clear-cut answer to the question what 'reasonably
large' is. Much depends on the contents of the box, but for the
sum of 100 draws the probability histogram will usually be very
close to the normal curve.
Don't confuse the number of draws and the number of
repetitions.
1
2
9
Example 1
Chapter 18
Chapter 18
Probability
Histograms
Probability
Histograms
Central Limit
Theorem
Central Limit
Theorem
Chapter 18
Example 2: solution to part 1
Probability
Histograms
Central Limit
Theorem
Chapter 18
Example 2
Roll a die 120 times
1 Use the normal
approximation to estimate
the chances of getting
between 15 and 25 sixes.
2 Use the normal
approximation to estimate
the chances of getting
exactly 20 sixes.
Example 2: solution to part 2
Probability
Histograms
Set up the box model: 120 draws with replacement from a box
with tickets 0,0,0,0,0,1.
Find EV and SE:
• EV = (# of draws) × (avg of box) = 120 × 61 = 20
• SE for sum of draws =
p
# of draws × (SD of the box)
• SD of box = (1 − 0) ×
q
1
6
• SE for sum of draws =
√
×
5
6
= .37
120 × .37 = 4.1
Normal approximation: New Average = 20, New SD = 4.1
#−Avg
25−20
z =
SD = 4.1 = 1.2
From normal table: Area=76.9%. So chance ≈ 76.9%
Central Limit
Theorem
What is EV and SE? Same as before! EV = 20, SD = 4.1
Find area under normal curve - but between what??
In the probability histogram, the bar over 20 goes from 19.5 to
20.5. So we should nd this area under the normal curve.
#−Avg
20.5−20
z =
SD = 4.1 = .12
From normal table: Area=8.0%. So chance ≈ 8.0%
Chapter 18
Continuity correction
Probability
Histograms
Central Limit
Theorem
Use this continuity correction whenever you're looking for an
exact set of numbers such as:
• Exactly 20: 19.5 to 20.5
• Between 15 and and 25, inclusive: 14.5 to 25.5
• Between 15 and and 25, exclusive: 15.5 to 24.5
• Between 15 and and 25: 15 to 25