class 15. the central limit theorem sprigg lane p 288
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Class 15. The Central Limit Theorem
Sprigg Lane
P 288
If you know and s
95% confidence interval for
𝑋𝑛∓𝑡 .𝑖𝑛𝑣 .2 𝑡(0.05 ,𝑑𝑜𝑓 )×𝑠√𝑛
There is a 95% pthis interval will cover μ.
Confidence Interval for the mean
Before Weights. Changing Counts Mean 82.36 82.36 82.36 82.36 82.36 82.36Standard Error 0.61 1.64 1.16 0.82 0.58 0.16Standard Deviation 5.184 5.18 5.18 5.18 5.18 5.18Sample Variance 26.875 26.87 26.87 26.87 26.87 26.87Count 72 10 20 40 80 1000t.inv.2t(.05,count-1) 1.99 2.26 2.09 2.02 1.99 1.96Confidence Level(95.0%) 1.218 3.708 2.426 1.658 1.154 0.322
Standard error goes down with
1/
2T inv t-value goes down as dof goes up…slowly.
Confidence interval gets
narrower with n.In this example, we kept sample mean and sample standard deviation constant.
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 p-values
• =norm.dist(X,μ,σ,true)• =norm.s.dist(Z,true)• =t.dist(T,dof,true)• =chisq.dist(chi2,dof,true)
• =t.dist.2t(T,dof)
• =t.dist.rt(T,dof)• =chidist(chi2,dof)• =chisq.dist.rt(chi2,dof)
The first four are LEFT TAIL
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 $82,142
The standard deviation is $77,430
The distribution is Weird and not normal
is a random variable
The same
$77,430/
Close to normal
The probability distribution of NPV
The probability distribution of 1/10th share of ten “identical” wells
Central Limit Theorem P 288
Implications of the CLT
In selecting simple random samples of size n from a population, the sampling distribution of the sample mean can be approximated by the normal distribution
as the sample size becomes large.
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).
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