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where &2 = SSRfull/(n - 5). In terms of the original indices, the full model is Y = {3o + {31X 1 + {32X2 + {33X3 + {34X:j + e and the reduced model is Y = {3o + {31X 1 + {32X2 + {34X:j + e. Assuming normality of the distribution of random errors, F has an F distribution with 1 and n - 5 degrees of freedom when {33 = o. The value of F for testing {33 = 0 in the turnip green data is 5.68 and the p-value is 0.03. Since the p-value is less than 0.05, most statisticians would reject the null hypothesis and conclude that the air temperature variable makes a significant contribution to the model. A Test Based on The value of should indicate whether {3J = 0 or not. Similar to the test statistic (3.5) for testing {3 = 0 in the case of simple regression,

131 Show the Huffman tree that results from the following distribution of punctuation characters and digits: colon (loo), space (605), newline ( 1 OO), comma (705), 0 (43 I), 1 (242), 2 ( 176), 3 (59), 4 (185), 5 (250), 6 (174), 7 (199), 8 (205), and 9 (217) 132 Most systems come with a compression program Compress several types of files to determine the typical compression rate on your system How large do the files have to be to make compression worthwhile Compare their performance with the Huffman coding program ( h z i p ) provided in the online source code 133 What happens if a file compressed with Huffman's algorithm is used to transmit data over a phone line and a single bit is accidentally lost What can be done in this situation

~ [u(kh"(l- y5)u(p)][u(p'h,,(1- y5)v(k')], (12.31)

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134 Prove the correctness of Huffman's algorithm by expanding the following steps a Show that no node has only one child b Show that the two least frequent characters must be the two deepest nodes in the tree c Show that the characters in any two nodes at the same depth can be swapped without affecting optimality d Use induction: As trees are merged, consider the new character set to be the characters in the tree roots

135 Under what circumstances could a Huffman tree of ASCII characters generate a 2-bit code for some character Under what circumstances could it generate a 20-bit code 136 Show that, if the symbols have already been sorted by frequency, Huffman's algorithm can be implemented in linear time 137 Huffman's algorithm occasionally generates compressed files that are larger than the original Prove that all compression algorithms must have this property (ie, no matter what compression algorithm you design, some input files must always exist for which the algorithm generates compressed files that are larger than the originals)

138 In the cross-reference generator, store the line numbers in a v e c t o r instead of a 1 i s t and compare performance 139 If a word occurs twice on a line, the cross-reference generator will list it twice Modify the algorithm so that duplicates are only listed once 1310 Modify the algorithm so that, if a word appears on consecutive lines, a range is indicated For example,

see (12.11), where the spinors are labeled by the particle momenta. Recall that the outgoing Pe is described by v (k '). The muon decay rate can now be obtained using (4.36), (12.32) where the invariant phase space is

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