Transcript Class 15. The Central Limit Theorem

Class 15. The Central Limit
P 288
Theorem
Sprigg Lane
Confidence Interval for the mean
If you know 𝑋𝑛 and s
There is a 95%
probability this interval
will cover μ.
95% confidence interval for μ
𝑠
𝑋𝑛 ∓ 𝑡. 𝑖𝑛𝑣. 2𝑡(0.05, 𝑑𝑜𝑓) ×
𝑛
Before Weights. Changing Counts
Mean
82.36
Standard Error
0.61
Standard Deviation
5.184
Sample Variance
26.875
Count
72
t.inv.2t(.05,count-1)
1.99
Confidence Level(95.0%)
1.218
Standard error
goes down with
1/ 𝑛
82.36
1.64
5.18
26.87
10
2.26
3.708
82.36
1.16
5.18
26.87
20
2.09
2.426
82.36
0.82
5.18
26.87
40
2.02
1.658
82.36
0.58
5.18
26.87
80
1.99
1.154
In this example, we kept sample mean and sample
standard deviation constant.
82.36
0.16
5.18
26.87
1000
1.96
0.322
2T inv t-value
goes down as dof
goes up…slowly.
Confidence
interval gets
narrower with n.
Hypothesis Tests
• Hypotheses about p’s
–
–
–
–
Binomial (she’s guessing)
Normal approximation when n is big (Wunderdog)
CHI-squared goodness of fit (Roulette Wheel)
CHI-squared independence (Supermarket Survey)
• Hypotheses about means
–
–
–
–
–
One-sample z-test (IQ μ=100 with σ=15)
One-sample t-test (IQ μ=100)
Two-sample t-test (heights μM = μF)
Two-sample paired t-test (Weight before and after)
ANOVA single factor (heights for three IT groups)
Using Excel function to calculate pvalues
•
•
•
•
=norm.dist(X,μ,σ,true)
=norm.s.dist(Z,true)
=t.dist(T,dof,true)
=chisq.dist(chi2,dof,true)
The first four are LEFT
TAIL
• =t.dist.2t(T,dof)
• =t.dist.rt(T,dof)
• =chidist(chi2,dof)
• =chisq.dist.rt(chi2,dof)
The last three are
RIGHT TAIL
Sprigg Lane
• Sprigg Lane is an Investment Company
• The Bailey Prospect is the site of a potential
well that has a 90% probability of natural gas.
• Federal Tax laws were recently changed to
encourage development of energy.
• The Bailey prospect will be packaged with 9
other similar wells
– Sprigg Lane plans to sell a large portion of the
package to outside investors.
Bailey Prospect Uncertainties
• Total Well Cost
– $160K +/- $5,400 (95% probability, normal)
• Enough Gas there to proceed?
– P=0.9
• Initial Amount in million cubic feet?
– lognormal(33,4.93)
• Btu content?
– 1055 to 1250 with 1160 most likely (BTU per cubic feet)
• Production Decline Rate multiplier
– .5 to 1.75 with 1 most likely
• Average Inflation (affecting costs and future gas prices)
– Normal(0.035,0.005)
Best-Guess Valuation
Analysis Agenda
• Analyze the riskiness of the baily prospect
project
– Replace each of the six uncertainties with a
probability distribution
– Find out the resulting probability distribution of
NPV.
• Analyze the riskiness of a 1/10th share of an
investment package of ten wells.
– This will be the distribution of a sample average of
ten NPVs.
Summary: The properties of the NPV
of the Bailey prospect
NPV is a random variable
The mean is
The standard deviation is
The distribution is
𝑋𝑛 is a random variable
$82,142
The same
$77,430
$77,430/ 10
Weird and not
normal
Close to normal
The probability distribution of NPV
The probability distribution of 1/10th
share of ten “identical” wells
Central Limit Theorem
P 288
In selecting simple random samples of size n from a population, the sampling
distribution of the sample mean 𝑋 can be approximated by the normal
distribution as the sample size becomes large.
Implications of the CLT
If the population (underlying probability distribution) is normal, our tests of
hypotheses about means WORK FINE.
If the population (underlying probability distribution) is NOT normal, our tests will
still work fine if n is big (>30 is a rule of thumb).