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Outline Chapter 6

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Schaum’s OutlinePROBABILTY and STATISTICS Chapter 6 ESTIMATION THEORYPresented by Professor Carol DahlExamples from D. Salvitti J. Mazumdar C. Valencia

- Trader in energy stocks
- random variable Y = value of share
- want estimates µy, σY
- Y = ß0 + ß1X+
- want estimates Ŷ, b0, b1
- Properties of estimators
- unbiased estimates
- efficient estimates

- Types of estimators
- Point estimates
- µ = 7
- Interval estimates
- µ = 7+/-2
- confidence interval

- Population parameters and confidence intervals Means
Large sample sizes

Small sample sizes

- Proportions
- Differences and Sums
- Variances
- Variances ratios

Unbiased Estimator of Population Parameter

estimator expected value = to population parameter

Population Parameters:

Sample Parameters:

are unbiased estimates

Expected value of standard deviation not unbiased

- Efficient Estimator –
if distributions of two statistics same

more efficient estimator = smaller variance

efficient = smallest variance of all unbiased estimators

Target

Estimates which are efficient and unbiased

Not always possible

often us biased and inefficient

easy to obtain

Point Estimate

single number

Interval Estimate

between two numbers.

- X = value of share
- sample mean is $32
- volatility is known σ2 = $4.00
- confidence interval for share value
- Need
- estimator for mean
- need statistic with
- mean of population
- estimator

- P(-1.96 < <1.96) = 95%

2.5%

- P(-1.96 < <1.96) = 95%
- P(-1.96 < <1.96 ) = 95%
- P(-1.96 -X < -µ <1.96 - X ) = 95%
- Change direction of inequality
- P(+1.96 +X > µ > -1.96 + X ) = 95%

- P(+1.96 +X > µ > -1.96 + X ) = 95%
- Rearrange
- P(X - 1.96 < µ <X + 1.96 ) = 95%
- Plug in sample values and drop probabilities
- X = value of share, sample = 64
- sample mean is $32
- volatility is σ2 = $4
- {32 – 1.96*2/64, 32 + 1.96*2/64} = {31.51,32.49}

- Take a sample
- point estimate
- compute sample mean
- interval estimate – 0.95 (95%+) = (1 - 0.05)
- X +/-1.96
- X +/-Zc
- (Z<Zc) = 0.975 = (1 – 0.05/2)
- 95% of intervals contain
- 5% of intervals do not contain

- interval estimate – 0.95 (95%+) = (1 - 0.05)
- X +/-Zc
- (Z<Zc) = 0.975 = (1 – 0.05/2)
- interval estimate – (1-) %
- X +/-Zc
- (Z<Zc) = (1 – /2)
- % of intervals don’t contain
- (1- )% of intervals do contain

(Z<Zc) = 0.975 = (1 – 0.05/2)

Common values for corresponding to various confidence levels used in practice are:

Functions in EXCEL

Menu Click on Insert Function or

=confidence(,stdev,n)

=confidence(0.05,2,64)= 0.49

X+/-confidence(0.05,2,64)

=normsinv(1-/2) gives Zc value

X+/-normsinv(1-/2)

32 +/- 1.96*2/64

Confidence interval

Confidence level

Evaluate density of oil in new reservoir

81 samples of oil (n)

from population of 500 different wells

samples density average is 29°API

standard deviation is known to be 9 °API

= 0.05

X = 29 , N= 500, n = 81 , σ = 9 , = 0.05

Zc = 1.96

But don’t know Variance

t-Distribution

=N(0.1)

=

= tdf

2/df

df

Confidence Intervals of Meanst- distribution

=

=

=

=

Confidence Intervals of MeansNormal compared to t- distribution

t distribution

Normal

X +/-Zc

X +/-tc

Example:

Eight independent measurements diameter of drill bit

3.236, 3.223, 3.242, 3.244, 3.228, 3.253, 3.253, 3.230

99% confidence interval for diameter of drill bit

X +/-tc

X +/-tc

X = ΣXi/n

3.236+3.223+3.242+3.244+3.228+3.253+3.253+3.230

8

X = 3.239

ŝ2 = Σ(Xi - X) = (3.236- X)2 + . . .(3.230 - X)2

(n-1) (8-1)

ŝ = 0.0113

X +/-tc

- X = 3.239, n = 8, ŝ= 0.0113, =0.01,
- 1- /2=0.995
- From the t-table with 7 degrees of freedom, we find tc= t7,0.995=3.50

1-/2=.975

.005%

-tc

tc

Find tc from Table of Excel

1-/2=.975

/2= 0.005%

Depends on Table

-tc

tc

GHJ /2 = 0.005 tc = 2. 499

Schaums 1- /2 = 0.995 tc = 2.35

Excel =tinv(0.01,7) = 3.499483

X +/-tc

X = 3.239, n = 8, ŝ= 0.0113, =0.01,

Example

600 engineers surveyed

250 in favor of drilling a second exploratory well

95% confidence interval for

proportion in favor of drilling the second well

Approximate by Normal in large samples

Solution: n=600, X=250 (successes), = 0.05

zc = 1.96 and

Example

600 engineers surveyed

250 in favor of drilling a second exploratory well.

95% confidence interval for

proportion in favor of drilling the second well

Approximate by Normal in large samples

Solution: n=600, X=250 (successes), = 0.05

zc = 1.96 and

sampling from large population

or finite onewith replacement

Samples are independent

Example

sample of 200 steel milling balls

average life of 350 days - standard deviation 25 days

new model strengthened with molybdenum

sample of 150 steel balls

average life of 250 days - standard deviation 50 days

samples independent

Find 95% confidence interval for difference μ1-μ2

Example

Solution: X1=350, σ1=25, n1=200, X2=250, σ2=50, n2=150

Where:

P1, P2 two sample proportions,

n1, n2 sizes of two samples

Example

random samples

200 drilled holes in mine 1, 150 found minerals

300 drilled holes in mine 2, 100 found minerals c

Construct 95% confidence interval difference in proportions

Solution: P1=150/200=0.75, n1=200, P2=100/300=0.33,n2=300

With 95% of confidence the difference of proportions {0.42, 0.08}

Example

Solution: P1=150/200=0.75, n1=200,

P2=100/300=0.33, n2=300

95% of confidence the difference of proportions

[0.08,0.42]

Need statistic with

population parameter 2

estimate for population parameter ŝ2

its distribution - 2

has a chi-squared distribution

n-1 degrees of freedom.

Find interval such that σ lies in the interval for

95% of samples

95% confidence interval

Rearrange

Take square root if want confidence interval for

standard deviation

Drop probabilities when substitute in sample values

1 - confidence interval for variance

1 - confidence interval for standard deviation

Example

Variance of amount of copper reserves

16 estimates chosen at random

ŝ2 = 2.4 thousand million tons

Find 99% confidence interval variance

Solution: ŝ2=2.4, n=16,

degrees of freedom = 16-1= 15

Not symmetric

/2

/2

2 lower 2 upper

1-/2

Not symmetric

1-/2

/2

/2

GHJ area above 20.995, 20.005 4.60092, 32.8013

Schaums area below 20.005, 20.995 4.60, 32.8

Excel = chiinv(0.995,15) = 4.60091559877155

Excel = chiinv(0.005,15) = 32.8013206461633

Example

99% confidence interval variance of reserves

Solution: ŝ=2.4 (n-1)=15

2lower = 4.60, 2upper = 32.8

Two independent random samples

size m and n

population variances

estimated variances ŝ21, ŝ22

- interested in whether variances are the same
- 21/ 22

Need statistic with

population parameter 21/ 22

estimate for population parameter ŝ21/ ŝ22

its distribution - F

F-Distribution

df1

df2

F-Distribution

Need statistic with

population parameter 21/ 22

estimate for population parameter ŝ21/ ŝ22

its distribution - F

Rearrange

Put smallest first, largest second

When substitute in values drop probabilities

1- confidence interval for 21/ 22

Example

Two nickel ore samples

of sizes 16 and 10

unbiased estimates of variances 24 and 18

Find 90% confidence limits for ratio of variances

Solution: ŝ21 = 24, n1 = 16, ŝ22 = 18, n2 = 10,

/2

/2

Tablesdf1df2

F upper

F lower

GHJ area above F0.95,15,9, F0.05,15,9 ?3.01 Schaums area below F0.05,15,9, F0.95,15,9 ?3.01

Area above

Excel = Finv(0.95,15,9) = 0.386454546279388

Excel = Finv(0.05,15,9) = 3.00610197251669

GHJ area above F0.95,15,9

P(F15,9>Fc) = 0.95

P(1/F15,9<1/Fc) = 0.95

But 1/F15,9 = F9,15

P(F9,15<1/Fc) = 0.95

P(F9,15<1/Fc) = 0.05

1/Fc = 2.59 Fc = 0.3861

/2

/2

F upper

F lower

Example

Two nickel ore samples

Solution: ŝ21 = 24, n1 = 16, ŝ22 = 18, n2 = 10,

Point Estimates

x is population with density function f(x,)

if know - know the density function

2 where = degrees of freedom

Poisson λxe-λ/x! = λ (the mean)

If sample independently from f n times

x1, x2, . . .xn

a sample

if consider all possible samples of n

a sampling distribution

If sample independently from f n times

x1, x2, . . .xn

a sample

if consider all possible samples of n

a sampling distribution

called likelihood function

which maximizes the likelihood function

Derivative of L with respect to and setting it to 0

Solve for

Usually easier to take logs first

log(L) = log(f(x1,) + log(f(x2,)+ . . .+ log(f(xn,)

log(L) = log(f(x1,) + log(f(x2,) +. . .+ log(f(xn,)

Solution of this equation is maximum likelihood estimator

work out example 6.25

work out example 6.26

- Y = ß0 + ß1X
- Ŷ, b0, b1
- Properties of estimators
- unbiased estimates
- efficient estimates
- Types of estimators
- Point estimates
- Interval estimates

- Y- µY, Y, Y, ŝ2
- In 590-690
- Y = ß0 + ß1X
- Ŷ, b0, b1
- Properties of estimators
- unbiased estimates
- efficient estimates
- Types of estimators
- Point estimates
- Interval estimates

- Need statistic with
- population parameter
- estimate for population parameter
- its distribution

- Population parameters and confidence intervals
- Mean – Normal
Know variance and population normal

Large sample size can use estimated variance

Proportions

- large sample approximate by normal
- Differences of means (known variance)

- Mean
- population normal - unknown variance

- Variances

- Variances ratios

- Maximum Likelihood Estimators
- Pick which maximizes the function