Explanation Given info: In an 2 4 experiment to study the power consumption on four factors each at two levels such as nature of roof, power setting, scrap used and charge. The treatment combinations were replicated three times for each combination of treatments. Calculation: Yates method is used to verify the sum of squares obtained from Minitab output. Yates method: Write the combination of treatments in the standard order in the beginning of the table called treatment column. Find the total effect obtained from each treatment combination and write it in the second column x i j k ⋅ . Find columns ( 1 ) , ( 2 ) , ... , ( n ) 1. Calculate column ( 1 ) by using the column x i j k ⋅ and the procedure is given below: 2. Upper half is obtained by adding the pairs of values in the column x i j k ⋅ 3. Lower half is obtained by taking the differences in pairs. That is, by subtracting the first value of pairs with the second value. 4. Similarly the columns ( 2 ) , ( 3 ) , ... , ( n ) are obtained by proceeding in the same manner as mentioned above. Finally, this process is repeated for n times in 2 n factorial experiment. Sum of squares due to contrast is obtained by using the formula given below: S S = ( contrast ) 2 48 The table below shows the Yates method of computation for the given data: Condition (1) x i j k l m (2) 1 (3) 2 (4) 3 (5) Effect contrast (6) S S = ( contrast ) 2 48 (7) ( 1 ) 2,528 5,114 9,612 20,181 39,371 ( a ) 2,586 4,498 10,569 19,190 387 3,120 ( b ) 2,354 5,872 9,354 –247 –3,993 332,168 ( a b ) 2,144 4,697 9,836 634 –845 14,876 ( c ) 2,961 5,264 –152 –1,791 1,439 43,140 ( a c ) 2,911 4,090 –95 –2,202 –193 776 ( b c ) 2,371 5,432 442 –263 –413 3,554 ( a b c ) 2,326 4,404 192 –582 295 1,813 ( d ) 2,446 58 –616 957 –991 20,460 ( a d ) 2,818 –210 –1,175 482 881 16,170 ( b d ) 2,010 –50 –1,174 57 –411 3,519 ( a b d ) 2,080 –45 –1,028 –250 –319 2,120 ( c d ) 2,598 372 –268 –559 –475 4,701 ( a c d ) 2,834 70 5 146 –307 1,964 ( b c d ) 2,224 236 –302 273 705 10,355 ( a b c d ) 2,180 –44 –280 22 –251 1,313 Error 16,4528 Where, The column x i j k ⋅ is obtained by adding the given values in pairs. That is, For ( 1 ) it is, 866 + 862 + 800 = 2 , 528 For ( a ) it is, 946 + 800 + 840 = 2 , 586 Similarly the remaining values are obtained. The column (3) named as 1 is calculated by using the following method: For ( 1 ) it is the addition of first two values from the column (2) named as x i j k ⋅ and it is shown below: 2 , 528 + 2 , 586 = 5 , 114 The same method is followed for the remaining 7 rows. The second half of column (3) named as 1is calculated by find the differences in pairs of values from the column (2) named as x i j k ⋅ 2 , 586 − 2 , 528 = 58 Then, 2 , 144 − 2 , 354 = − 210 The fourth column named as 2, fifth column named as 3 and sixth column named as effect contrast were also calculated by using the method as given above. Finally, the S S = ( contrast ) 2 48 is calculated by taking the square of Effect contrast and then dividing it by 48 since n is three. The general formula for SS is S S = ( contrast ) 2 16 ( n ) , the denominator n is multiplied by a factor 16 because there are sixteen treatment combinations in 2 4 experiment and n is three since there are three observations for each treatment combinations. Total sum of squares is calculated as follows: SST = ∑ ∑ ∑ ∑ ∑ x i j k l m 2 − ( x ⋅ ⋅ ⋅ ⋅ ⋅ ) 2 16 n Substitute ∑ ∑ ∑ ∑ ∑ x i j k l m 2 as 32,917817, n as 3 and x ⋅ ⋅ ⋅ ⋅ ⋅ is 39 , 371 ( = 2 , 528 + 2 , 586 +

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