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Example 17 - Mean of 5 observations is 4.4, variance is 8.24 - Indirect questions - finding remaining observations

Example 17 - Chapter 15 Class 11 Statistics - Part 2
Example 17 - Chapter 15 Class 11 Statistics - Part 3 Example 17 - Chapter 15 Class 11 Statistics - Part 4 Example 17 - Chapter 15 Class 11 Statistics - Part 5

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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. ๐‘†๐‘ข๐‘š ๐‘œ๐‘“ ๐‘œ๐‘๐‘ ๐‘’๐‘Ÿ๐‘ฃ๐‘Ž๐‘ก๐‘–๐‘œ๐‘›๐‘ ๏ทฎ๐‘๐‘ข๐‘š๐‘๐‘’๐‘Ÿ ๐‘œ๐‘“ ๐‘œ๐‘๐‘ ๐‘’๐‘Ÿ๐‘ฃ๐‘Ž๐‘ก๐‘–๐‘œ๐‘›๐‘ ๏ทฏ = 4.4 1 + 2 + 6 + ๐‘ฅ + ๐‘ฆ๏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( ๐‘ฅ๏ทฎ๐‘–๏ทฏ๏ทฏโˆ’ ๐‘ฅ๏ทฏ)๏ทฎ2๏ทฏ = 8.24 1๏ทฎ5๏ทฏ ๏ทฎ๏ทฎ( ๐‘ฅ๏ทฎ๐‘–๏ทฏ๏ทฏโˆ’4.4)๏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 [ (โ€“3.4)2 + (โ€“2.4)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๏ทฎ๐‘ฆ๏ทฏ Putting (3) in (1) x + y = 13 36๏ทฎ๐‘ฆ๏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๏ทฎ๐‘ฆ๏ทฏ = 36๏ทฎ4๏ทฏ = 9 Hence x = 9, y = 4 are the remaining two observations For y = 9 x = 36๏ทฎ๐‘ฆ๏ทฏ = 36๏ทฎ9๏ทฏ = 4 Hence, x = 4, y = 9 are the remaining two observations Thus, remaining observations are 4 & 9

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.