Transcript document
The Guessing Game or What’s an Educated Guess? Do not worry, this is not a presentation on the benefits of ESP! The “guess” referred to in the title is not a wild stab, but a reasoned, reasonable estimate of what some unknown number could be, including (eventually) a reasoned, reasonable estimate of the chance of our estimating incorrectly. Let’s review the situation: There is a given population with unknown population mean , usually unknown population standard deviation , and maybe some unknown parameter p you would also like to know. The population is huge, or maybe your testing is destructive, or prohibitively expensive, so you sample, that is … … you pick at random n elements of the population, get n numbers x1, x2, x3, …, xn apply some arithmetic to the numbers and get a new number (a statistic) (g for guess, “hat” for tradition!) Note the following two facts: 1. The formula you use on the n numbers you got can be as wild as you please (a free country!) 2. Probabilities stay the same in each pick and probabilities multiply Various Educated Guesses Let me give you six methods you could use to guess the population mean : 1. The sample average : x 2. The Bidding Selection Method BS Throw away the highest and the lowest numbers, average the rest, 3. The cheapie method CM: Always pick the second number you got 4. The “glass is half-empty” method gL gL = the lowest number you got. 5. The “glass is half-full” method gH gH = the highest number you got. 6. Granny’s method GM: GM = the best looking number you got. OK, let’s discard Granny’s method, it’s very subjective! But what about the other five, are there any you would discard as bad guesses for the population mean, and why? Let’s list them: x,BS, CM, gL, gH So, are there any of these statistics (that is what they are, numbers associated to samples!) you would not use as reasonable guesses for the population mean? The trouble is ….. … we have not decided what reasonable means! Maybe we’ll have better luck deciding what unreasonable means! (without night there would be no day, concepts are often defined by their limitations or opposites) For example I submit to you that gL and gH are unreasonable guesses for the population mean. Why? Let’s do an example. Our population consists of: 1,000 beanbags, some weighing we get the following 27 possible samples, each with its “half-empty glass” statistic and corresponding probability This new statistic has three values 0, 2, 7, and its probability distribution is (gotten by adding the probabilities in the previous table.) Remember the wonderful secret No such luck this time, we get instead E(gL) = 2x0.216 + 7x0.027 = 0.432 + 0.189 = 0.621 far below the actual value of ! That’s right, the statistic gL underestimates, because on the average it gives a value much lower than what it is supposed to estimate. Similarly (check it out!) the statistic gH overestimates, because on the average it gives a value much higher than what it is supposed to estimate. So what statistic Y are we looking for? Obviously one that neither underestimates nor overestimates, that is one that on the average hits it on the nose. This means that E(Y) = the parameter to be estimated. Any statistic Y that neither underestimates nor overestimates the population parameter we are trying to estimate (that is E(Y) = ) is called an UNBIASED point estimator The word “point” here means we are guessing one value (one point on the real line) as opposed to guessing an interval as we will do later. MVUE’ s Let me let you in on a little secret: The three point estimators of : x , BS, CM we met before are all three unbiased. So … which is the best of the three? or more precisely how do we compare them? (i.e., what does best mean?) Quick Answer: Best means the one (if it exists) that gives us the highest probability of being correct. Recall than variance measures how wildly spread the values can be. The smaller the variance the more tightly packed the values are (around their expected value). Therefore between two unbiased estimators we pick the one with smaller variance. Between a million unbiased estimators we pick the one with the smallest variance (if it exists!) and call it a M(inimum) V(ariance) U(nbiased) E(stimator) The power of the mean Fact: Among (the millions and millions of) all possible unbiased estimators of the population mean , the sample mean is the MVUE. (for once our intuition is correct !) The weakness of the Variance Recall that when we defined the sample variance we added a “correction factor” n/(n-1) that is we wrote The correction is due to the fact that without it the sample variance would underestimate the population variance. The correction makes s2 unbiased, but raises its own variance. The choices are: 1. On the nose but a little wilder 2. Minimum wildness but slightly off (No best guess this time, it does not exist!) We will often need to guess the population variance and, in the interest of correctness, we will choose Option 1. Actually, if n is large, either one is acceptable.