Estimation and Uncertainty

12-706/73-359Lecture 8 - Sept 24, 2003

Problem of Unknown Numbers

If we need a piece of data, we can: Look it up in a reference source Collect number through survey/investigation Guess it ourselves Get experts to help you guess it

Often only ‘ballpark’, ‘back of the envelope’ or ‘order of magnitude needed Situations when actual number is unavailable or

where rough estimates are good enough E.g. 100s, 1000s, … (102, 103, etc.)

Source: Mosteller handout

Notes about Reference Sources

Some obvious: Statistical Abstract of US Always check sources and secondary

sources of data Usually found in footnotes – also tells you

about assumptions/conditions for using Sometimes the summarized data is wrong!

Look in multiple sources Different answers implies something about

the data and method – and uncertainty

Estimation in the Course

We will encounter estimation problems in sections on demand, cost and risks.

We will encounter estimation problems in several case studies.

Projects will likely have estimation problems.

Need to make quick, “back-of-the-envelope” estimates in many cases. Don’t be afraid to do so!

Estimation gets no respect

The 2 extremes - and the respect thing Aristotle:

“It is the mark of an instructed mind to rest satisfied with the degree of precision which the nature of the subject permits and not to seek an exactness where only an approximation of the truth is possible.”

Archbishop Ussher of Ireland, 1658 AD: “God created the world in 4028 BC on the 9th of

September at nine o’clock in the morning.”

We consider it somewhere in between

In the absence of “Real Data”

Are there similar or related values that we know or can guess? (proxies) Mosteller: registered voters and population

Are there ‘rules of thumb’ in the area? E.g. ‘Rule of 72’ for compound interest r*t = 72: investment at 6% doubles in 12 yrs MEANS construction manual

Set up a ‘model’ to estimate the unknown Linear, product, etc functional forms Divide and conquer

Methods

Similarity – do we have data that can be made applicable to our problem?

Stratification – segment the population into subgroups, estimate each group

Triangulation – create models with different approaches and compare results

Convolution – use probability or weightings (see Selvidge’s table, Mosteller p. 181) Note – example of a ‘secondary source’!!

Notes on Estimation

Move from abstract to concrete, identifying assumptions

Draw from experience and basic data sources

Use statistical techniques/surveys if needed Be creative, BUT Be logical and able to justify Find answer, then learn from it. Apply a reasonableness test

Attributes of Good Assumptions

Need to document assumptions in course Have some basis in known facts or

experience Are unbiased towards the answer Example: what is inflation rate next year?

Is past inflation a good predictor? Can I find current inflation? Should I assume change from current

conditions? We typically use history to guide us

How many TV sets in the US?

Can this be calculated? Estimation approach #1:

Survey/similarity How many TV sets owned by class? Scale up by number of people in the

US Should we consider the class a

representative sample? Why not?

TV Sets in US – another way

Estimation approach # 2 (segmenting): work from # households and # tvs per

household - may survey for one input Assume x households in US Assume z segments of ownership (i.e.

what % owns 0, owns 1, etc) Then estimated number of television

sets in US = x*(4z5+3z4+2z3+1z2+0z1)

TV Sets in US – sample

Estimation approach # 2 (segmenting): work from # households and # tvs per

household - may survey for one input Assume 50,000,000 households in US Assume 19% have 4, 30% 3, 35% 2,

15% 1, 1% 0 television sets Then

50,000,000*(4*.19+3*.3+2*.35+.15) = 125.5 M television sets

TV Sets in US – still another way

Estimation approach #3 – published data

Source: Statistical Abstract of US Gives many basic statistics such as

population, areas, etc. Done by accountants/economists - hard

to find ‘mass of construction materials’ or ‘tons of lead production’.

How close are we?

How well did we do?

Most recent data = 1997 But ‘recently’ increasing < 3% per year

TV/HH - 125.5 tvs, StatAb – 229M tvs, % error: (229M – 125.5M)/125.5M ~

82% What assumptions are crucial in

determining our answer? Were we right? What other data on this table validate our

models?

Significant Figures

We estimated 125,500,000 tvs in USHow accurate is this - nearest 50,000,

the nearest 500,000, the nearest 5,000,000 or the nearest 50,000,000?

Should only report estimates to your confidence - perhaps 1 or 2 “significant figures” could be reported here.

Figures are only carried along to document calculations or avoid rounding errors.

Some handy/often used data

Population of US btw 275-300 millionNumber of households ~ 100 millionAverage personal income ~$30,000

Estimate Annual Vehicle Miles Travelled (VMT) in the US

Estimate “How many miles per year are passenger automobiles driven in the US?”

Types of models Similar to TVs: Guess number of cars,

segment population into miles driven per year

Find fuel consumption data, guess at fuel economy ratio for passenger vehicles

Other ideas? Let’s try it on the board.

Estimate VMT in the US

Table 1033 of Stat. Abstract suggests 1995 VMT was 2.068 trillion miles (yes - twice as much as 1972 implied in the Mosteller handout)! 230 billion ‘passenger car trips’ per year About 200 million cars Avg VMT 21,000 mi.

More clever: Cobblers in the US

Cobblers repair shoesOn average, assume 20 min/taskThus 20 jobs / day ~ 5000/yr

How many jobs are needed overall for US?I get shoes fixed once every 5 years

About 280M people in USThus 280M/4 = 56 M shoes fixed/year

56M/5000 ~ 11,000 => 10^4 cobblers in USActual: Census dept says 5,120 in US

An Energy Example

Energy measured in SI units = Watts (as opposed to BTUs, etc)

In practice, we usually talk about kilowatts or kilowatt-hours of energy

Rule: 1 Watt of energy used for one hour is One watt-hour (compound unit) = 1Wh 1000 Watts used for one hour = 1kWh

‘How much energy used by lighting in US residences?’

‘How much energy used by lighting in US residences?’

Assume 50 light fixtures per houseAssume each in use avg 2 hours per dayAssume average fixture is 50WThus each fixture uses 100Wh/dayEach house uses 5000Wh/day (5kWh/day)100 million households would use 500

million kWh/day 182,500 million kWh/yr

‘How much energy used by lighting in US residences?’

Our guess: 182,500 million kWh/yr DOE: “lighting is 5-10% of household elec” http://www.eren.doe.gov/erec/factsheets/eelight.html

2000 US residential Demand ~ 1.2 million million kWh (source below) 10% is 120,000 million kWh 5% is 60,000 million kWh 2000 demand source:

http://www.eia.doe.gov/cneaf/electricity/epm/ epmt44p1.html

A Random Example

Select a random panel of data from the Statistical Abstract of the U.S. Can you formulate an ‘estimation

question’? Can you estimate the answer? How close were you to the ‘actual

answer’?Let’s try this ourselves

Uncertainty

Investment planning and benefit/cost analysis is fraught with uncertainties forecasts of future are highly uncertain applications often made to preliminary

designs data is often unavailable

Statistics has confidence intervals – economists need them, too.