The article “A Thin-Film Oxygen Uptake Test for the Evaluation of Automotive Crankcase Lubricants” (?Lubric. Engr., ?1984: 75–83) reported the following data on oxidation-induction time (min) for various commercial oils: a?? alculate the sample variance and standard deviation. b. ?If the observations were reexpressed in hours, what would be the resulting values of the sample variance and sample standard deviation? Answer without actually performing the reexpression.

Answer : Step 1 : Given The article “A Thin-Film Oxygen Uptake Test for the Evaluation of Automotive Crankcase Lubricants”. Consider the data 87 145 103 153 130 152 160 138 180 87 195 99 132 93 145 119 211 129 105 Now we have to calculate the sample variance and standard deviation. 2 x xi 87 7569 103 10609 130 16900 160 25600 180 32400 195 38025 132 17424 145 21025 211 44521 105 11025 145 21025 153 23409 152 23104 138 19044 87 7569 99 9801 93 8649 119 14161 129 16641 x =2563 x =368501 i i So we have to find mean. The formula of the mean is xi x = n Substitute the value x = 20,179 and n=27. i 2,563 x = 19 x = 134.8947 The formula of the sample variance is. 2 ( x ) 2 (xi) n s = n1 Substitute the value i , n and x. ( 2563 ) 2 (368501) 19 s = 191 ( 658969) 2 (368501) 19 s = 18 s = (368501) 345735.2105 18 2 (22765.78947 s = 18 s = 1,264.7660 Therefore sample variance is 1,264.7660 Then we have to find standard deviation. Standard deviation is square root of the sample variance. We know that sample variance is 1,264.7660. s = 1,264.7660 s = 35.5635 Therefore the standard deviation is 35.5635.