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LECTURE 16 EX P2 - ‘42 Chapter 4 I Randomized Blocks...

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Unformatted text preview: ‘42 Chapter 4 I Randomized Blocks, Latin Squares, and Related Designs ITABLE 4.11 Analysis of Variance for the Rocket Propellant Experiment W Source of Sum of Degrees of Mean Variation Squares Freedom Square F0 P«Vaiue Fomulations 330.00 4 82.50 _ 7.73 0.0025 Batches of raw material 68.00 4 17.00 Operators 150.00 4 37.50 Error 128.00 12 10.67 Total 676.00 24 As in any design problem, the experimenter should investigate the adequacy of the model by inspecting and plotting the residuals. For a Latin square, the residuals are given by eijk : yijk _ 217k _ _ _ =m—n—m—m+h- The reader should find the residuals for Example 4.3 and construct appropriate plots. A Latin square in which the first row and column consists of the letters written in alpha- betical order is called a standard Latin square, which is the design used in Example 4.4. A standard Latin square can always be obtained by writing the first row in alphabetical order and then writing each successive row as the row of letters just above shifted one place to the left. Table 4.12 summarizes several important facts about Latin squares and standard Latin squares. As with any experimental design, the observations in the Latin square should be taken in random order. The proper randomization procedure is to select the particular square employed at random. As we see in Table 4.12, there are a large number of Latin squares of a particular size, so it is impossible to enumerate all the squares and select one randomly. The I TA B L E 4 . 1 2 ltandard Latin Squares and Number of Latin Squares of Various Sizes“ iize 3X3 4X4 5X5 6X6 7X7 po ixamplesof ABC ABCD ABCDE ABCDEF ABCDEFG ABC.r.P standardsquares BCA BCDA BAECD BCFADE BCDEFGA BCD...A CAB CDAB CDAEB CFBEAD CDEFGAB CDE...B DABC DEBAC DEABFC DEFGABC ECDBA EADFCB EFGABCD FDECBA FGABCDE PAB...(P—l) GABCDEF lumber of l 4 56 9408 16,942,080 —— standard squares “otal number of 12 576 161,280 818,851,200 61,479,419,904,000 p!(p— 1)1>< Latin squares (number of standard squares) w Some of the information in this table is found in Fisher and Yates (1953). Little is known about the properties of Latin squares larger than 7 X 7. ...
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