1. Estimation and Uncertainty 12-706/ 19-702 Lecture 2

2. Announcements / Etc. zToday’s slides posted after class zNo class on Friday, Monday zHW 1 Handed Out zOffice Hours: Mon, Tues 430-530 yIn CEE alumni lounge (118) y1 TA in non-HW weeks, 2 TA when hw

3. 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!

4. 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, … (10 2, 10 3, etc.)  Source: Mosteller handout

5. 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

6. 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

7. 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

8. Methods zSimilarity – do we have data that can be made applicable to our problem? zStratification – segment the population into subgroups, estimate each group zTriangulation – create models with different approaches and compare results zConvolution – use probability or weightings (see Selvidge’s table, Mosteller p. 181) yNote – example of a ‘secondary source’!!

9. 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

10. Attributes of Good Assumptions  Need to document assumptions in course  Write them out and cite your sources  Have some basis in known facts or experience  Write why you make the specific assumptions  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

11. 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?

12. TV Sets in US – another way  Estimation approach # 2 (segmenting):  Work from # households and # TV’s 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*(4z 5 +3z 4 +2z 3 +1z 2 +0z 1 )

13. 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% have 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

14. 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?

16. How well did we do?  Most recent data = 2004  But ‘recently’ increasing < 2% per year  TVs - 125.5 tvs, StatAb – 248M TVs,  % error: (248M – 125.5M)/125.5M ~ 100%  What assumptions are crucial in determining our answer? Were we right?  What other data on this table validate our models?  See ‘SAMPLE ESTIMATION’ linked on web page to see how you are expected to answer these types of questions.  Also see “SAMPLE SPREADSHEET” for a suggested organization in Excel

17. Notes on Sample Estimation Files zThe text file gives the type and structure of documentation we expect when doing assumption-based analysis. There is a question like it on Homework 1, make sure your answer looks like that. zThe spreadsheet file suggests a framework for building assumptions into spreadsheets, i.e., placing them all at the top where you can see them. If needed, you can use the cell values as links in your equations. zNote the Excel plug-ins we will use later will want to see assumptions done like this.

18. Changing Assumptions zStatistical Abstract gave additional info: yAverage TVs/HH = 2.4 (ours was 2.5) yNumber of households: 100 million (ours 50) zThus to redo our analysis, we should do a better job at estimating households

19. Significant Figures zWe estimated 125,500,000 TVs in US zHow accurate is this - nearest 50,000, the nearest 500,000, the nearest 5,000,000 or the nearest 50,000,000? zShould only report estimates to your confidence - perhaps 1 or 2 “significant figures” could be reported here. zFigures are only carried along to document calculations or avoid rounding errors.

20. Notes on References - Check and Double Check Sources (and dates) zTop 3 google sites for “US population”. y281,421,906 (factmonster.com “2000”)  302,510,402 (wikipedia, census - July 2007) y302,724,715 (census pop. Clock - live) zNumber of households y108 million (2005, US census) zUS avg personal income: $36,300

21. Avoiding Point Estimates zThe tradeoff in this kind of work is getting away with a guess yAnd giving an informed-enough answer that doesn’t sound like a guess! zReally what we should be doing is making ranges of estimates yWe will refer to these as lower bound, mean, and upper bound estimates yYou might think of lower bound as “5th percentile” and upper as “95th percentile” ySo they’re not true lower/upper bounds (which might be zero and infinity).

22. Uncertainty zInvestment planning and benefit/cost analysis is fraught with uncertainties yforecasts of future are highly uncertain yapplications often made to preliminary designs ydata is often unavailable zStatistics has confidence intervals – we need them, too zWe will talk in more detail about uncertainty in a few weeks.

23. Exercise #2: Estimate Annual Vehicle Miles Travelled (VMT) in the US zEstimate “How many miles per year are passenger automobiles driven in the US?” zTypes of models ySimilar to TVs: Guess number of cars, segment population into miles driven per year yFind fuel consumption data, guess at fuel economy ratio for passenger vehicles yOther ideas? Let’s try it on the board.

24. Estimate VMT in the US zTable 1084 of 2006 Stat. Abstract suggests 2003 VMT was 2.7 trillion miles (yes - twice as much as 1972 implied in the Mosteller handout)! yAbout 200 million cars yabout 12,000 miles per car zNote the Dept of Transportation separately specifies “passenger car VMT” as 1.7 trillion miles - does better job of separating trucks yAbout 16k VMT per household yhttp://www.bts.gov/publications/national_transportat ion_statistics/2006/index.html (Table 1-32)http://www.bts.gov/publications/national_transportat ion_statistics/2006/index.html

25. More clever: Cobblers in the US zCobblers repair shoes

26. More clever: Cobblers in the US zCobblers repair shoes zOn average, assume 20 min/task zThus 20 jobs / day ~ 5000/yr yHow many jobs are needed overall for US? zI get shoes fixed once every 5 years yAbout 280M people in US zThus 280M/4 = 56 M shoes fixed/year y56M/5000 ~ 11,000 => 10^4 cobblers in US zActual: Census dept says 5,120 in US

27. A Random Example zSelect a random panel of data from the Statistical Abstract of the U.S. (1998) ySomething not likely to have changed much yCan you formulate an ‘estimation question’? yCan you estimate the answer? yHow close were you to the ‘actual answer’? zLet’s try this ourselves

28. Form Small Groups zMake groups of 3-4 zPick one of the problems on the handout and work on it for 5-10 minutes zFinish for HW 1 (group)