risk in agriculture paul d. mitchell aae 575. goal how to make economically optimal...
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Risk in Agriculture
Paul D. MitchellAAE 575
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Goal How to make economically optimal
decisions/choices under risk First: Discuss how to talk about risk
(measuring risk) Second: How to make economically
optimal choices under risk (risk management)
Like before: First production functions, Second economics
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What is Risk? Risk is a four-letter word!!!
“Possibility of a loss” “Chance of a bad outcome”
Risk versus Uncertainty Risk: outcomes and probabilities are not
know Soybean Rust in USA; Ag Bioterrorist Attack
Uncertainty: outcomes and probabilities are known CBOT corn price for next fall; Crop yield
distributions A fine distinction that many people ignore
and use the two terms interchangeably Technically, we will do Uncertainty, but call it
Risk
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Major Categories of Agricultural Risk
1. Production and Technical Risk2. Market and Price Risk3. Financial Risk4. Human Resource Risk5. Legal and Institutional Risk
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Production and Technical Risk
Uncertainty in crop yields or livestock gains due to numerous factors Weather: flood, drought, hail, frost, etc. Pests and Diseases: ECB, CRW, Soybean
Aphid, Soybean Rust, BSE, brucellosis, etc.
New Technologies: new herbicides, hybrids (transgenics), tillage, planter, harvest machines, milking facilities, organic, intensive grazing methods, IPM, soil testing, etc.
Input Shortages: labor, custom machinery or application, trucking, fungicides (for rust)
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Market and Price RiskUncertainty in market prices or in
ability to market production Input price changes: fuel, fertilizer,
fungicide, feed/grain, etc. Crop and livestock prices vary
continuously (CBOT, CME, etc.) Market Access: Hurricane Katrina and
barge traffic shut down last fall Processor/Contractor/Buyer: goes out of
business or changes quality requirements
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Financial Risk
Money borrowed or external equity provided creates risk Interest rate changes Change in value of assets used as
collateral Ability to generate income to meet debt
obligations (liquidity and solvency) Lender’s/investor’s willingness to
continue lending/providing capital
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Human Resource RiskSeveral people are key to a farm business
and potential for changes creates risk Employee management problems: retention,
turnover, criminal activity, disputes, etc. Injury, illness, death of manager/key employee Key employee, spouse, child: retires, career
change, relocates, etc. Family disputes, divorces, etc.: personal stress,
plus losses from legal settlements, property diversions, financial reallocations, etc.
Estate Planning: how are farm assets going to be transferred between generations? Can create risk
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Legal and Institutional Risk
Created by regulations and legal liabilities Regulations for manure, chemicals,
facility siting, antibiotic use, carcass disposal, burning
Liability for accidents: machinery, livestock
Labor laws: taxes, worker health and safety, residency requirements
Contractual obligations: hedge-to-arrive and futures contracts, contracts with processors
Tax liability: properly file all required forms
Ignorance of law is not a legal excuse
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Risk is a Major Topic
We will just barely touch the surface Our Focus: Production/technical risk
Calculating and interpreting commonly used measures of risk (How to measure risk)
Applying common criteria for decision making under uncertainty (How to manage risk)
Common tools to manage different risks Measurement first, so can describe
effect of risk management
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Measuring Risk
Convert “Risk” to “Uncertainty” Know all outcomes and their probabilities
Gives the “probability density function”
Statistical function describing all possible outcomes and associated probabilities Discrete: die roll, coin flip, game Continuous: tomorrow’s max
temperature, corn price at harvest, earnings next year
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Discrete Example: Probability Density Function of Course Grade for Two Persons
Course Grade(outcome)
Person 1(probability)
Person 2(probability)
A 0.70 0.05
AB 0.15 0.15
B 0.10 0.40
BC 0.05 0.20
C 0.00 0.10
D 0.00 0.05
F 0.00 0.05
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
A AB B BC C D F
Course Grade
Pro
bab
ility
Person 1
Person 2
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Continuous DistributionMaximum temperature t has a normal
distribution with a mean of 70 and standard deviation of 10, so its probability density function is:
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
30 50 70 90 110
temperature
pro
ba
bili
ty d
en
sit
y
2
22
1 ( 70)~ ( ) exp
102 10
tt f t
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Many Types of Distributions Many possible pdf’s, discrete and
continuous, some with upper and/or lower limits, some symmetric or skewed, etc.
Lognormal (prices) Beta (yields, % losses, rates of gain) Gamma or Weibull (yields) Triangular (low information) Empirical: draw from data (discrete)
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Measures of Risk Goal: have a function (pdf):
outcomes and probabilities How do you describe this pdf? Typically focus on measures of 1) Central tendency or location 2) Spread or variability 3) Skewness or “bad” outcomes
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Measures of Risk Mean/Expected Value/Average Median Mode Standard Deviation Variance Coefficient of Variation Confidence Interval Probability of Key Events (i.e., p < 0)
Central Tendency or Location
Spread or Variability
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Mean Average or expected outcome Probability weighted average of random
variable Discrete: If x is a random variable with N
possible outcome values, each with probability pi, then the mean or expected value of x is
Continuous: If x is a random variable with probability density function f(x), then the mean or expected value of x is:
1
E[ ]N
i ii
x p x
E[ ] ( )x
x xf x dx
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Simple Example A crop has three possible yields:
low 50 bu/A, with probability 0.25 typical 100 bu/A, with probability
0.60 high 150 bu/A, with probability
0.15 Expected Yield = Mean = m = 0.25 x 50 + 0.60 x 100 + 0.15 x 150
= 12.5 + 60 + 22.5 = 95 bu/A Mean is a “probability weighted
average”
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Another Example Profit for a crop has 4 possible outcomes
with probabilities as reported in table Mean = 50 + 5400 + 3200 + 1800 =
$10,450Yiel
dPric
eProbabili
ty Profit Probability x Profit
lo lo 0.05 $1,000 $50
lo hi 0.45$12,00
0 $5,400
hi lo 0.40 $8,000 $3,200
hi hi 0.10$18,00
0 $1,800
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Interpreting Means The mean is not what will happen, but rather,
if the random event occurs several times, the mean is the average of the outcomes The mean die roll is 3.5, which does not imply that
if you roll a die you will get a 3.5. Rather if you roll a die several times, the average of all these rolls will be close to 3.5
If your mean corn yields is 150 bu/ac, this does not imply that next year you will get 150 bu/ac, rather the average of your corn yields over the next several years will be around 150 bu/ac (if you plant the same hybrids)
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Central Tendency
Besides Mean, the Median & Mode also measure a distribution’s Central Tendency
Median = the middle or half way point Half of the draws will be < the median Half of the draws will be > the median
Mode = most common or most likely value
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Mean-Median-Mode
Symmetric Distribution: Mean = Median Mean = Median = Mode: Normal
Distribution Mean = Median ≠ Mode: Uniform (die
roll) Skewed/Asymmetric Distribution
Mean ≠ Median ≠ Mode Gamma, Beta, Lognormal, Weibull, etc.
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Mean ≠ Median ≠ Mode
0.00.51.01.52.02.53.03.5
0 0.2 0.4 0.6 0.8 1
% Survival per Spray
pro
bab
ility
den
sity
Beta distribution for % survival per spray of ECB larvae in sweet corn after 1, 3, and 5 sprays of Capture
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Measures of Spread or Variability
Probability weighted measures of variability or spread of possible outcomes
All begin with the deviation of the possible outcomes from the mean
If xi is an outcome and E[x] = mean, then Deviation Di = xi – E[x] = xi – m
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Variance
Variance = s2 =
Variance = probability weighted average of the squared deviation of each outcome from the mean
Variance = sort of the average squared deviation
Why squared deviation? Converts all deviations to be positive Negative and positive deviations of same size
have same value: –32 = 9 and 32 = 9
2 2
1 1
( E[ ])N N
i i i ii i
p D p x x
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Standard Deviation
Standard Deviation = square root of theVariance:
Why take the square root of the variance? Variance units are squared unit of Mean Standard Deviation units are same as Mean’s If yield has Mean of bu/a, then the Variance is
bu2/A2 and the Standard Deviation is bu/A Think of it as the “typical deviation from mean” Technically “square root of the mean squared
deviation”
2
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Calculating Variance and St. Dev.
Previous Example: three possible yields: Low 50, Typical 100, High 150 Probabilities 0.25, 0.60, and 0.15 Mean is 95 bu/A
D12 = (50 – 95)2 = 452 = 2025
D22 = (100 – 95)2 = 52 = 25
D32 = (150 – 95)2 = 552 = 3025
Var. = 2025 + 25 + 3025 = 5075 bu2/A2
St. Dev. = sqrt(5075) = 71.24 bu/A
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Variance and St. Dev. Example
Previous Table: 4 possible profits, Mean = $10,450 Variance = 4,465,125 + 1,081,125 + 2,401,000 +
5,700,250 Variance = 13,647,500 squared dollars Standard Deviation = sqrt(13,647,500) = $3,694
Yield
Price
Prob. Profit
Squared Deviation
Prob. X Squared
Deviation
lo lo 0.05 $1,00089,302,5
00 4,465,125
lo hi 0.45$12,00
02,402,50
0 1,081,125
hi lo 0.40 $8,0006,002,50
0 2,401,000
hi hi 0.10$18,00
057,002,5
00 5,700,250
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0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140
pro
ba
bili
ty d
en
sit
y
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140
pro
ba
bili
ty d
en
sit
y
Normal Density with st. dev. of 15 and means of 50 (red)70 (green)90 (blue)
Normal Density with mean of 70 and st. dev. of 10 (red)15 (green)20 (blue)
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0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
pro
ba
bili
ty d
en
sit
yBeta Density with mean of 0.30 and st. dev. of 0.10 (red)0.15 (green)0.20 (blue)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
pro
ba
bili
ty d
en
sit
y
Beta Density with st. dev. of 0.15 and means of 0.25 (red)0.30 (green)0.35 (blue)
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Coefficient of Variation Coefficient of Variation (CV) is s/m
(st. dev./mean), expressed as a percent
If standard deviation is 60 and mean 200, then CV = 60/200 = 0.30 or 30%
CV Normalizes the standard deviation by the mean, thus expressing the standard deviation as a percentage of the mean
CV of 30% implies the standard deviation is 30% of the mean
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Coefficient of Variation (CV) CV is a relative measure of risk CV “corrects for” difference in mean Corn price and yield (Coble, Heifner and Zuniga
2000) Price variability the same everywhere (CV 21%-
23%) Yield variability varies greatly by location (CV
15%-40%)-------- Price -------- -------- Yield --------
County meanst.
dev. CV meanst.
dev. CV
Iroquois, IL 2.24 0.49 22% 143.2 28.6 20%
Douglas, KS 2.33 0.49 21% 90.48 30.8 34%
Lincoln, NE 2.20 0.46 21% 150.3 22.5 15%
Pitt, NC 2.52 0.58 23% 87.7 35.1 40%
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How Actually Done
Discrete pdf: 50 bu Pr = 0.25, 100 bu Pr = 0.60, 150 bu Pr = 0.15 Calculate
Continuous pdf: don’t do integrals yourself, use formulas
People get confused by normal pdf where parameters = mean & st dev
Most pdfs are not this easy
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Some pdfs
Lognormal, Beta, Gamma, Weibull Look up the pdf and formula's for the
mean and st dev etc. in books, Wikipedia, Wolfram alpha, Google, etc.
Main Point: lots of pdfs out there, mean and variance are not the parameters, but functions of the parameters
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Lognormal pdf
Pdf: Min = 0 Max = +infinity Mean = exp(m + ½s2) Variance = exp(2m)exp(s2)(exp(s2) –
1) If y ~ lognormal, then ln(y) ~ normal
with mean m and st dev s
2
22
1 (ln( ) )~ ( | , ) exp
22
yy f y
y
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Beta pdf
Pdf: Min = 0 Max = 1 Mean = a/( a + w) Variance = aw/[( a + w)2( a + w + 1)] Can re-scale to be between other
upper and lower limits
1 1(1 )~ ( | , )
( , )
y yy f y
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Beta pdf
Pdf:
Beta Function:
Gamma Function:
Note: Programs have the gamma function
1 1(1 )~ ( | , )
( , )
y yy f y
1
1 1
0
( , ) (1 )u u du ( ) ( )
( , )( )
1
0
( ) exp( ) zz u u du
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Rescaled Beta pdf
y ~ beta(a,w) with min 0 and max 1 z = L + (U – L)y also has a beta
distribution, but with Min = L Max = U Mean = L + (U – L)(a/( a + w)) Variance = (U –
L)2(aw/[(a+w)2(a+w+1)])
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Rescaled Beta pdf
If y ~ beta(a,w), then z = L + (U – L)y has a pdf
1 1
1
( ) ( )~ ( | , , , )
( , )( )
z L U zz f z L U
U L
1 1
1
( ) ( ) ( ) ( )~ ( | , , , )
( )( )
z L U zz f z L U
U L
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Gamma pdf
Pdf: Min = 0 Max = +infinity Mean = bl Variance = b2l
1( / ) exp( / )~ ( | , )
( )
y yy f y
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Weibull pdf
Pdf: Min = 0 Max = +infinity Mean = bG[(l + 1)/l] Variance =
b2G[(l + 2)/l] – b2(G[(l + 1)/l])2
1
~ ( | , ) expy y
y f y
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Confidence Interval
Confidence Interval: the limits between which the random variable will be with a predefined probability
Example: 95% confidence interval for corn yield: Corn Yield is between 100 bu/ac and 195 bu/ac with 95% probability
Another measure of variability: a wider confidence interval implies greater variability
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Confidence Interval Rule of Thumb
95% confidence interval is approximately the mean plus and minus 2 standard deviation
65% confidence interval is approximately the mean plus and minus 1 standard deviation
Example: Suppose returns have a mean of $100/ac and a standard deviation of $25/ac, then
Returns in range $50-$150/ac with 95% probability
Returns in range $75-$125/ac with 65% probability
Approximation close for most random variables
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Confidence Intervals
Use the inverse of the cumulative distribution function (CDF) to calculate confidence intervals
If y ~ f(y), then F(z) = Pr(y ≤ z) f(y) is the pdf and F(y) is the cdf 95% CI
F(zlo) = 0.025, so F-1(0.025) = zlo
F(zhi) = 0.975, so F-1(0.975) = zhi
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Probability of Key Events
Sometimes key events important What’s the probability that:
Yield is less than 80 bu/ac? Price is less than $1.90/bu? Returns will exceed $5/ac?
Break Even Probability: probability that Recover the investment cost? Get the cost of insecticide back in saved
yield? Per acre returns will be positive?
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Probability of Key Events: If y ~ f(y), then Pr(y ≤ z) =
F(z) What’s the probability that:
Yield is less than 80 bu/ac? y ~ f(y), then Pr(y < 80) = F(80)
Price is less than $1.90/bu? p ~ f(p), then Pr(p < 1.90) = F(1.90)
Returns will exceed $5/ac? r ~ f(r), then Pr(r > 5) = 1 – F(5)
Break Even Probability: probability that Per acre returns will be positive?
p ~ f(p), then Pr(p > 0) = 1 – F(0)
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Probabilities
In special cases, can calculate probability of key events, but usually need numerical simulations (Monte Carlo analysis)
Confidence Interval: pick the probability and then derive the limits
Probability of Key Events: pick the limit or limits and the derive the probability
Value at risk (VAR): pick the probability and the limit, find the portfolio allocation to get them
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Cumulative Distribution Functions and Their Inverses If z ~ f(z) and F(z) is CDF, a = Pr(z ≤ Z) =
F(Z) Standard Normal CDF: a = F(z)
Excel NORMDIST(Z, 0, 1, true) If ~ N(m,s) then CDF: = a F((Z – m)/s)
Excel NORMDIST(Z, m, s, true) Inverse Standard Normal CDF z = F-1(a)
Excel NORMSINV(a,0,1) If ~ N(m,s) then inverse CDF z = F-1(a) + s
m Excel NORMSINV(a,m,s)
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Normal CDF: m = 30, s = 10a = Pr(z ≤ Z) = F(Z)a = F(Z)Non-standard normala = F((Z – m)/s)a = NORMDIST(Z,m,s,true)
z = F-1(a)Non-standard normalz = F-1(a) + s mz = NORMINV(a,m,s)
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Cumulative Distribution Functions and Their Inverses Many CDF’s require special functions to
evaluate Some software packages have them, some
don’t Lognormal CDF: F(z) = F{(ln(z)-m)/s} Lognormal inverse CDF: F-1(a) = exp[F-
1(a)s+m] Weibull CDF: F(z) = 1 – exp[–(z/b)l] Weibull inverse CDF: F-1(z) = [b ln(1/(1-a))] 1-l
Beta CDF: incomplete beta function (now Excel)
Gamma CDF: ????
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Some pdfs
Lognormal, Beta, Gamma, Weibull Look up the cdf and formula's for the
mean and st dev etc. in books, Wikipedia, Wolfram alpha, Google, etc.
Main Point: lots of pdfs out there, mean and variance are not the parameters, but functions of the parameters
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Extended Example
Bt Corn Yield Simulations with local ECB pressure and yield parameters
Wisconsin State Average ECB and Yield
Hall County, NE (irrigation with lots ECB)
Random Variables: yield, ECB pressure, ECB tunneling, % yield loss
10,000 Monte Carlo random draws
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0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0 25 50 75 100 125 150 175 200 225 250
Yield bu/ac
Pro
bab
ilit
y D
ensi
ty
Bt
No Bt
0.000
0.002
0.004
0.006
0.008
0.010
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0 25 50 75 100 125 150 175 200 225 250
Yield bu/ac
Pro
bab
ilit
y D
ensi
ty
Bt
No Bt
Wisconsin
Hall County, NE
Simulated empirical probability density functions of harvested yield with and without Bt corn in Wisconsin and Hall County, NE
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Bt HallNo Bt
Hall Bt WI No Bt WI
Mean 168.2 155.6 130.2 124.4
Median 172.1 158.9 133.5 127.2
Mode* 190 165 155 150
Variance 1140.6 1144.6 1531.9 1448.3
St Dev 33.77 33.83 39.14 38.06
CV 20.1% 21.7% 30.1% 30.6%• Bt corn increases mean yield in both locations,
more in Hall County where more pest pressure exists
• Bt corn decreases variance and standard deviation in Hall County and increases both in WI (irrigated vs dryland)
• Bt corn increases yield CV in both locations, more in Hall County (risk measure matters)
* Rounded to nearest 5 bu
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Bt HallNo Bt
Hall Bt WI No Bt WI
50% lo 147.1 133.9 103.7 98.6
50% hi 193.8 180.8 160.6 153.1
70% lo 131.6 118.7 86.2 81.8
70% hi 203.4 190.9 172.6 165.7
90% lo 105.5 94.6 59.6 56.5
90% hi 216.8 206.0 188.9 182.2
Different confidence intervals for Bt and non-Bt corn in both Hall County and Wisconsin
See graphics to better understand differences
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0.000
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0 25 50 75 100 125 150 175 200 225 250
0.000
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0.000
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0 25 50 75 100 125 150 175 200 225 250
50% Confidence Interval: Upper limit further from mean
70% Confidence Interval: Limits approximately symmetric around mean90% Confidence Interval: Lower limit further from mean
Pattern holds for Bt and non-Bt in both locations
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Example ContinuedBreak-Even Probability
What’s the probability, given an expected price and yield, that you will recover the Bt corn “Tech Fee” in saved yield?
Analyzed this question for WI regions UWEX bulletin and Spreadsheet on
class web page