DECS 430-A Business Analytics I: Class 5 Sampling Polling, and estimating proportions Choosing a sample size Sampling methods Stratified sampling, cluster sampling Sampling problems Non-response bias, measurement bias Optimization (Excels Solver) Adverse selection Polling If the individuals in the population differ in some
qualitative way, we often wish to estimate the proportion / fraction / percentage of the population with some given property. For example: We track the sex of purchasers of our product, and find that, across 400 recent purchasers, 240 were female. What do we estimate to be the proportion of all purchasers who are female, and how much do we trust our estimate? First, the Estimate Let 240 p 0.6 60% .
400 Obviously, this will be our estimate for the population proportion. But how much can this estimate be trusted? And Now, the Trick Imagine that each woman is represented by a 1, and each man by a 0. Then the proportion (of the sample or population) which is female is just the mean of these numeric values, and so estimating a proportion is just a special case of what weve already done!
The Result Estimating a mean: s x (~ 2) n Estimating a proportion: p(1 - p) p (~ 2) n [When all of the numeric values are either 0 or 1, s takes the special form shown above.]
0.6(1 - 0.6) , or 60% 4.8% . The example: 0.6 (~ 2) 400 Multiple-Choice Questions If the Republican Partys candidate were to be chosen today, which one would you most prefer? Romney, Cain, Bachman, Perry, Gingrich, Santorum, Paul, Huntsman, none The results are reported as if 9 separate yes/no questions had been asked. If the Republican Partys candidate were to be chosen today, which of these would have your approval?
The same reporting method is used. Choice of Sample Size Set a target margin of error for your estimate, based on your judgment as to how small will be small enough for those who will be using the estimate to make decisions. Theres no magic formula here, even though this is a very important choice: Too large, and your study is useless; too small, and youre wasting money. Estimating a Proportion: Polling Pick the target margin of error. Why do news organizations always use 3% or
4% during the election season? Because thats the largest they can get away with. p(1 - p) 0.5(1 - 0.5) 1 (~ 2) (~ 2) n n n So, for example, n=400 (resp., 625, or 1112) assures a margin of error of no more than 5% (resp., 4%, or 3%).
Estimating a Mean: Choice of Sample Size Set the target margin of error. s Solve (~ 2) t arg et n target = $25. s $180. Set n = 207. From whence comes s? From historical data (previous studies) or from a pilot study (small initial survey). The Square-Root Effect : Choice of Sample
Size after an Initial Study Given the results of a study, to cut the margin of error in half requires roughly 4 times the original sample size. And generally, the sample size required to achieve a desired margin of error = 2 original margin of error original sample size desired m arg in of error How to Read Presidential-Race Polls When reading political polls, remember that
the margin of error in an estimate of the gap between the two leading candidates is roughly twice as large as the poll's reported margin of error. The margin of error in the estimated change in the gap from one poll to the next is nearly three times as large as the poll's reported margin of error. Summary Whenever you give an estimate or prediction to someone, or accept an estimate or prediction from someone, in order to facilitate risk analysis be sure the estimate is accompanied by its margin of error: A95%-confidence interval is (one standard-deviations-worth of uncertainty
(your estimate) (~2) inherent in the way the estimate was made) If youre estimating a mean using simple random sampling: s x (~ 2) n If youre estimating a proportion using simple random sampling: p (~ 2) p(1 - p) n How Will the Data be Collected? Primary Goals: No bias
High precision Low cost Simple random sampling with replacement Typically implemented via systematic sampling Simple random sampling without replacement Typically done if a population list is available Stratified sampling Done if the population consists of subgroups with relative within-group homogeneity
Cluster sampling Done if the population consists of (typically geographic) subgroups with substantial within-group heterogeneity Specialized approaches (e.g., tagging the U-Haul fleet) Non-Response Bias One of the difficulties in surveying people (whether by mail, telephone, or direct approach) is that some choose not to respond. Assume that you have decided to conduct a study which requires a sample size of 100. If you only expect 10% of those surveyed to respond to your questionnaire, what should you do? A nave answer is, "Simply send out 1000 questionnaires!" Unfortunately, the demographics of respondents and
nonrespondents may differ substantially. To base estimates for the entire population merely on the data collected from respondents therefore might leave you exposed to substantial sampling bias. Non-Response Bias A form of stratified sampling is typically used to overcome non-response bias. An initial mass mailing of questionnaires takes place, with identifying codes placed on each questionnaire (or its return envelope). When the submission deadline for responses is reached, estimates can be made for the stratum of "people who respond to the initial mailing." Crossing these people off the mailing list (by cross-referencing the codes on their responses) leaves a list of people all of whom are now known to be in the other "people who don't respond" stratum. The initial response rate is used to estimate the relative sizes of the two strata.
A sample of those who didn't respond is now recontacted, using a more expensive approach designed to obtain responses from everyone. (The expense is typically related to an incentive of some kind.) Their data provides estimates for the second stratum, and the study can then be completed. See Nonresponse_Bias.xls for an example. Measurement Bias Asking sensitive questions Software piracy Sexual activities Tax fraud People will lie Allow them to hide behind a mask of
randomness Randomized Response Surveys inverted response innocuous response Flip two coins: If both are tails, answer the following question untruthfully; otherwise, answer the queston truthfully. Flip two coins: If at least one is a head, go to A; otherwise, go to B. A: Flip a coin. Have you ever shoplifted? B: Flip a coin. Did you get a head?
Have you ever shoplifted? 75% Pr(answer actual question) 50% flip 1,000 sample size "Y" rate 55.00% 58.08% 51.92% 3.08% 60.00% estimate 66.17% 95%-confidence
64.09% 95%-confidence 55.91% limits 4.09% margin of error Larger samples are required for the same precision But the bias can be completely eliminated. See Sampling.xls for details. Optimization Using Excels Solver add-in Take My Car. Please!
Have I got a deal for you! I've got this great used car, and I might be willing to sell. The actual value of the car depends on how well it has been maintained, and this is of course only known to me: Expressed in terms of the car's value to me, you believe it to be equally likely to be worth any amount between $0 and $5000. You, who would utilize the car to a greater extent than I, would derive 50% more value from ownership (e.g., if it's worth $3000 to me, then it's worth $4500 to you). How much are you willing to offer me? (I'll interpret your offer as "take-it-or-leave-it.") Adverse Selection You are subject to adverse selection whenever 1. You offer to engage in a transaction with another
party, and that party can either accept or refuse your offer. 2. The other party holds information not yet available to you concerning the value to you of the transaction. 3. The other party is most likely to accept the offer (i.e., to select to do the deal) when the information is "bad news" (i.e., adverse) to you. Adverse Selection: Dealing with It We need to be able to compute E[ V | V v] . For normally-distributed uncertainty, this can be done analytically. (See Adverse_Selection_plus.xls) Adverse Selection: Examples
Making a buyout offer Setting an insurance premium getting (forcing) healthy young people to carry insurance is critical to the ACA Giving bid/ask quotes Auctions with objective value uncertainty contracting (unknown costs) natural resource sales (unknown supply) the Winners Curse debt auctions (unknown post-auction market price) Heres another Saturday night mothers teach daughters to avoid giving bad signals
Course Finale Weve covered Enough probability to get you started in FINC-430, OPNS-430, and other courses dealing with risk. Enough statistics to begin DECS-431, on regression analysis. Enough warning to provide a bit of protection against common errors. Good luck, and bon voyage!
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