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).