Chapter 4 Approximating a Sampling Distribution At the end of the last chapter we saw how tedious it is to find the sampling distribution of U even when there are only 20 possible assignments We also experienced the limit of my comfort zone 252 possible assignments For studies like Dawn s 184 756 possible assignments and especially Sara s 1 075 1023 possible assignments there are way too many possible assignments to seek an exact answer Fortunately there is an extremely simple way to obtain a good approximation subject to the caveats given below to a sampling distribution regardless of how large the number of possible assignments 4 1 Two Computer Simulation Experiments Let s return to Dawn s study Our goal is to create a table for Dawn that is analogous to Table 3 7 on page 65 for Kymn s study i e we want to determine the value of u x y for every one of the 184 756 possible assignments This is too big of a job for me Instead of looking at all possible assignments we look at some of them We do this with a computer simulation experiment I wrote a computer program that selected 10 000 assignments for Dawn s study For each selection the program selected one assignment at random from the collection of 184 756 possible assignments You can visualize this as using our randomizer website 10 000 times For each of the 10 000 simulated assignments I determined its value of u x y My results are summarized in Table 4 1 Suppose that we want to know P U 0 By definition it is the proportion of the 184 756 possible assignments that would yield u 0 We do not know this proportion because we have not looked at all possible assignments But we have with the help of my computer looked at 10 000 assignments of these 10 000 assignments 732 gave u 0 see Table 4 1 The relative frequency of u 0 in the assignments we have examined is an intuitively obvious approximation to the relative frequency of u 0 among all possible assignments The relative frequency of u 0 among all possible assignments is by definition P U 0 To summarize our approximation of the unknown P U 0 is the relative frequency of assignments that gave u 0 which is from our table 0 0732 The above argument for P U 0 can be extended to P U u for any of the possible 75 Table 4 1 The results of a 10 000 run simulation experiment for Dawn s study u Freq 3 6 1 3 4 1 3 2 3 3 0 4 2 8 16 2 6 25 2 4 45 2 2 78 2 0 113 1 8 191 1 6 240 1 4 335 Relative Freq 0 0001 0 0001 0 0003 0 0004 0 0016 0 0025 0 0045 0 0078 0 0113 0 0191 0 0240 0 0335 u Freq 1 2 419 1 0 506 0 8 552 0 6 662 0 4 717 0 2 729 0 0 732 0 2 765 0 4 716 0 6 674 0 8 553 1 0 507 Relative Freq 0 0419 0 0506 0 0552 0 0662 0 0717 0 0729 0 0732 0 0765 0 0716 0 0674 0 0553 0 0507 u Freq 1 2 394 1 4 315 1 6 251 1 8 150 2 0 108 2 2 93 2 4 54 2 6 23 2 8 17 3 0 8 3 2 3 Relative Freq 0 0394 0 0315 0 0251 0 0150 0 0108 0 0093 0 0054 0 0023 0 0017 0 0008 0 0003 Table 4 2 Selected probabilities of interest for Dawn s CRD and their approximations Probability of Interest P U 2 2 Its Approximation r f U 2 2 0 0198 P U 2 2 r f U 2 2 1 0 0105 0 9895 P U 2 2 r f U 2 2 0 0198 0 0173 0 0371 values u But it will actually be more interesting to us to approximate probabilities of more complicated events than P U u In particular recall that Dawn s actual u was 2 2 As we will see in Chapter 5 we will be interested in one or more of the probabilities given in Table 4 2 Let me give you some details on how the answers in this table were obtained To obtain r f U 2 2 we must sum the relative frequencies for the values 2 2 2 4 2 6 2 8 3 0 and 3 2 From Table 4 1 we obtain 0 0093 0 0054 0 0023 0 0017 0 0008 0 0003 0 0198 For r f U 2 2 note that this value is 1 r f U 2 2 1 0 0105 Finally r f U 2 2 is the sum of two relative frequencies U 2 2 and U 2 2 The first of these has been found to equal 0 0198 The second of these is 0 0078 0 0045 0 0025 0 0016 0 0004 0 0003 0 0001 0 0001 0 0173 76 Table 4 3 Selected probabilities of interest for Sara s CRD and their approximations Probability of Interest Its Approximation P U 8 700 r f U 8 700 0 0903 P U 8 700 r f U 8 700 0 9107 P U 8 700 r f U 8 700 0 0903 0 0921 0 1824 Adding these we get r f U 2 2 r f U 2 2 0 0198 0 0173 0 0371 Next I performed a computer simulation experiment for Sara s CRD As stated earlier there are more than 1023 different assignments for a balanced study with n 80 total trials Trying to enumerate all of these would be ridiculous so we will use a computer simulation with 10 000 runs My simulation study yielded 723 distinct values of u This is way too many to present in a table as I did for Dawn s study The simulation study for Dawn recall yielded 35 distinct values for u and that was unwieldy Recall that for Sara s data x 106 875 and y 98 175 giving u 8 700 Table 4 3 presents information that will be needed in Chapter 5 4 2 How Good are These Approximations Tables 4 2 and 4 3 present six unknown probabilities and their respective approximations based on simulation experiments with 10 000 runs Lacking knowledge of the exact probabilities I cannot say exactly how good any of these approximations are What I can say however is that each of them is very likely to be very close to the exact unknown probability it is approximating How can I know this Well we will see how later in this course when we learn about confidence intervals so you will need to be patient And of course the terms very likely and very close are quite vague Here is what we will do for now First the expression very close will …
View Full Document
Unlocking...