To raise money for a new rectory, the members of a church hold a raffle. A total
ID: 3172522 • Letter: T
Question
To raise money for a new rectory, the members of a church hold a raffle. A total of n tickets are sold (numbered 1 through n), out of which a total of fifty winners are to be drawn presumably at random. The following are the fifty lucky numbers. Set up a goodness-of-fit test that focuses on the randomness of the draw. Use the 0.05 level of significance.
108 110 21 6 44 89 68 50 13 63 8464 69 92 12 46 78 113 104 105 115 58 220 19 96 28 72 81 32 75 349 86 94 61 35 31 56 17100 102 114 76 106 112 80 59 73 3250-6663 610288577 6324229149 6 13 92 04 2 2 4315 69 38833 02 80 1093883520 256152 308 10 68 6 7 5 6 7 61 12 0848565102 1667 9760 8886993 94 17 06 8946992476 08 13910Explanation / Answer
chronological order of the data Flagging (A if above Median; B if Below Median) U : Runs 108 A 1 89 A 84 A 46 B 2 9 B 19 B 32 B 94 A 3 17 B 4 106 A 5 110 A 68 B 6 64 B 78 A 7 115 A 96 A 75 A 61 B 8 100 A 9 112 A 21 B 10 50 B 69 A 11 113 A 58 B 12 28 B 3 B 35 B 102 A 13 80 A 6 B 14 13 B 92 A 15 104 A 2 B 16 72 A 17 49 B 18 31 B 114 A 19 59 B 20 44 B 63 B 12 B 105 A 21 20 B 22 81 A 23 86 A 56 B 24 76 A 25 73 A Median 68.5 Total runs 25 Randomness test for Runs has been used to test the randomness of the data Step 1: Data has been considered in the chronological order Step 2: Flag has been created. If given observation is above Median then it is flagged as A otherwise B. Step 3: Runs has been calculated. Run is defined as a sequence of letters of one kind surrounded by a sequence of letters of one kind. These are presented in last column named as U Step 4: Calculation of Mean and Variance Mean: E(U) = n+2/2 E(U)=50+2/2 E(U) 26 Variance: V(U)=n/4*{n-2/n-1} V(U)=50/4*{50-2/50-1} V(U) 12.24489796 Step 5: Since n (which is 50 in this case ) is large , U may be regarded as asymptotically normal . Hence we use normal test (Z-test) at 0.05 level of significane to test for randomness Z Test: Null hypothesis: The winners has been drawn randomly Alternate Hypothesis: The winners has not been drawn randomly Test statistic Z=(U-E(U)) / sqrt(V(U)) Z -0.28577378 Level of Significance is given at 0.05 Critical or Rejection region at 0.05 level of significance is below -1.645 and +1.645 Non Critical or non rejection region at 0.05 level of significance is between -1.645 and +1.645 Conclusion: If the Z-statistic value fall in the non critical or non rejection region then we do not reject the null hypothesis Here the calculated Z-statistic vale -0.2858 that falls between -1.645 and +1.645 which is the non rejection region. Hence we do not have enough evidence to reject the claim (null hypothesis) that the winners has been drawn randomly Note Please check if the data I have considered is actually in chronological order or not . Because this could effect the results As the data has been given in tabular form ,I was not sure about the order in which they occurred. Hence I assumed column by column as the chronological order
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