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BUAD 300Dr. R. RobinsonNonparametric Test for RunsIs it possible to judge the randomness of observed data on the basis of the order observed? Can we test whether patterns that appear non-random are due to chance? For example, can we judge whether “a run of good or bad luck” is purely due to random chance? The so called nonparametric test for runs allows us to formally test patterns for chance. It is based upon a simple binomial (above or below, heads or tails) probability test.A run is a succession of identical letters (or any other binomial symbols) which is followed or preceded by a different set of letters. Consider the following time-sequenced arrangement of defective (indicated by d) and non-defective (indicated by o) pieces produced in a given order.ooooo ddddd oooooooooo dd oo dddd o dd ooAs observed, at first there is a run of 5 non-defective pieces, followed by a run of 5 defective pieces, followed by 10 non-defective, and so forth. There are 9 total runs out of 33 observations. Is this too few to be random? Is it possible to have too many runs to be considered random? Allow the following symbols:u = total # of runs no = # of letters ond = # of letters dThis is actually a simple problem in binomial probability, but as you know, if the binomial trials are sufficiently large in number, the binomial probabilities can be approximated by the normal. If the sequence of letters d and o are random, then E(u) and var(u) are given below by (1) and (2). Equation (3) is the standard-normal z, and is usable if no and nd are both 10 or more. The variable u is the actual (observed number of runs).E(u) = 2nondno+ nd+ 1(1)Var(u) = 2nond2nond - no- nd(no+ nd)2(no+ nd- 1)(2)z = u-E(u)√Var(u) (3)Since we could have either too few or too many runs, i.e., the actual u could be either greater than or less than E(u), then we reject the hypothesis of randomness if zcalc. ≥ zα/2 . We have a two tailed test. Question: Are the runs indicated in the problem above random?The solution to this problem (below) is in your Power Point presentation.Level of significance:no = nd = E(u) = 2nondno+ nd+ 1 = Var(u) = 2nond2nond - no- nd(no+ nd)2(no+ nd- 1) = z = u-E(u)√Var(u) = Decision rule: If zcalc. < zα/2 then do not reject H0.Decision concerning hypothesis of randomness: For the following assignment you are to estimate model equation (1) below, and when you do this use the “Storage” button to save your “standardized residuals.” These will be analyzed by the “runs analysis” of the material above. To do this, use the “runs test” command in Minitab under the “Nonparametric” list. You will find this under “Stat.”Coin Int = β0 + β1 Free Playt + ut (1) Assignment: Estimate (1) above via Minitab and replicate the runs analysis output in your Power Point presentation.Are the errors associated with your OLS estimates for equation (1) random? Use Minitab for these calculations. A run will be a series of errors above or below the mean of 0. Note that the p-value in Minitab is the probability that the errors are random given the number of calculated runs. Include your Minitab printout of your runs analysis for answering this. Also include a printout of your data and standardized errors. (Send this to me via email.) You must be able to interpret this Minitab printout, i.e. the expected number of runs, the actual number of runs, and the probability (p-value) that your errors are random. ................
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