



Examples
Example 2
Example 3 Important
Example 4
Example 5 Important
Example 6 (Normal Method)
Example 6 (Shortcut Method) Important
Example 7 Important
Example 8
Example 9 Important
Example 10 Important
Example 11
Example 12 Important
Example 13 Deleted for CBSE Board 2022 Exams
Example 14 Important
Example 15 Important Deleted for CBSE Board 2022 Exams
Example 16
Example 17 Important You are here
Example 18
Example 19 Important
Examples
Last updated at May 29, 2018 by Teachoo
Example 17 The mean of 5 observations is 4.4 and their variance is 8.24. If three of the observations are 1, 2 and 6, find the other two observations. Let the other two observations be x and y. Therefore, our observations are 1, 2, 6, x, y. Given Mean = 4.4 i.e. ๐๐ข๐ ๐๐ ๐๐๐ ๐๐๐ฃ๐๐ก๐๐๐๐ ๏ทฎ๐๐ข๐๐๐๐ ๐๐ ๐๐๐ ๐๐๐ฃ๐๐ก๐๐๐๐ ๏ทฏ = 4.4 1 + 2 + 6 + ๐ฅ + ๐ฆ๏ทฎ5๏ทฏ = 4.4 9 + x + y = 4.4 ร 5 x + y = 22 โ 9 x + y = 13 Also, Given Variance = 8.24 1๏ทฎn๏ทฏ ๏ทฎ๏ทฎ( ๐ฅ๏ทฎ๐๏ทฏ๏ทฏโ ๐ฅ๏ทฏ)๏ทฎ2๏ทฏ = 8.24 1๏ทฎ5๏ทฏ ๏ทฎ๏ทฎ( ๐ฅ๏ทฎ๐๏ทฏ๏ทฏโ4.4)๏ทฎ2๏ทฏ = 8.24 1๏ทฎ5๏ทฏ [ (1 โ 4.4)2 + (2 โ 4.4)2 + (6 โ 4.4)2 + (x โ 4.4)2 + (y โ 4.4)2 ] = 8.24 1๏ทฎ5๏ทฏ [ (โ3.4)2 + (โ2.4)2 + (1.6)2 + (x โ 4.4)2 + (y โ 4.4)2 ] = 8.24 1๏ทฎ5๏ทฏ [11.56 + 5.76 + 2.56 + x2 + (4.4)2 - 2(4.4)x + y2 + (4.4)2 - 2(4.4)y] = 8.24 [ 19.88 + x2 + 19.36 โ 8.8x + y2 + 19.36 โ 8.8y] = 8.24 ร 5 [ 19.88 + 19.36 + 19.36 + x2 + y2 โ 8.8y โ 8.8x ] = 41.2 [ 58.6 + x2 + y2 โ 8.8(x + y) ] = 41.2 [ 58.6 + x2 + y2 โ 8.8(13) ] = 41.2 58.6 + x2 + y2 โ 114.4 = 41.2 x2 + y2 = 114.4 + 41.2 โ 58.6 x2 + y2 = 97 From (1) x + y = 13 Squaring both sides (x + y)2 = 132 x2 + y2 + 2xy = 169 97 + 2xy = 169 2xy = 169 โ 97 2xy = 72 xy = 1๏ทฎ2๏ทฏ ร 72 xy = 36 x = 36๏ทฎ๐ฆ๏ทฏ Putting (3) in (1) x + y = 13 36๏ทฎ๐ฆ๏ทฏ + y = 13 36 + y(y) = 13(y) 36 + y2 = 13y y2 โ 13y + 36 = 0 y2 โ 9y โ 4y + 36 = 0 y(y โ 9) โ 4(y โ 9) = 0 (y โ 4)(y โ 9) = 0 So, y = 4 & y = 9 For y = 4 x = 36๏ทฎ๐ฆ๏ทฏ = 36๏ทฎ4๏ทฏ = 9 Hence x = 9, y = 4 are the remaining two observations For y = 9 x = 36๏ทฎ๐ฆ๏ทฏ = 36๏ทฎ9๏ทฏ = 4 Hence, x = 4, y = 9 are the remaining two observations Thus, remaining observations are 4 & 9