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Misc 2 - Mean, variance of 7 observations are 8, 16. If five - Miscellaneous

Misc 2 - Chapter 15 Class 11 Statistics - Part 2
Misc 2 - Chapter 15 Class 11 Statistics - Part 3 Misc 2 - Chapter 15 Class 11 Statistics - Part 4 Misc 2 - Chapter 15 Class 11 Statistics - Part 5

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Misc 2 The mean and variance of 7 observations are 8 and 16, respectively. If five of the observations are 2, 4, 10, 12 and 14. Find the remaining two observations. Let the other two observations be x and y. Therefore, our observations are 2, 4, 10, 12, 14, x, y. Given Mean = 8 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 = 8 2 + 4 + 10 + 12 + 14 + ๐‘ฅ + ๐‘ฆ๏ทฎ7๏ทฏ = 8 42 + x + y = 7 ร— 8 x + y = 56 โ€“ 42 x + y = 14 Also, Given Variance = 16 1๏ทฎn๏ทฏ ๏ทฎ๏ทฎ( ๐‘ฅ๏ทฎ๐‘–๏ทฏ๏ทฏโˆ’ ๐‘ฅ๏ทฏ)๏ทฎ2๏ทฏ = 16 1๏ทฎ7๏ทฏ ๏ทฎ๏ทฎ( ๐‘ฅ๏ทฎ๐‘–๏ทฏ๏ทฏโˆ’8)๏ทฎ2๏ทฏ = 16 1๏ทฎ7๏ทฏ [(2 โ€“ 8)2 +(4 โ€“ 8)2+(10 โ€“ 8)2+(12 โ€“ 8)2 +(14 โ€“ 8)2+(x โ€“ 8)2 +(y โ€“ 8)2] = 16 1๏ทฎ7๏ทฏ [ (โ€“6)2 + (โ€“4)2 + (2)2 + (4)2 + (6)2 + (x โ€“ 8)2 + (y โ€“ 8)2 ] = 16 1๏ทฎ7๏ทฏ [36 + 16 + 4 + 16 + 36 + x2 + (8)2 - 2(8)x + y2 + (8)2 - 2(8)y] = 16 [ 108 + x2 + 64 โ€“ 16x + y2 + 64 โ€“ 16y] = 16 ร— 7 [ 236 + x2 + y2 โ€“ 16y โ€“ 16x ] = 112 [ 236 + x2 + y2 โ€“ 16(x + y) ] = 112 [ 236 + x2 + y2 โ€“ 16(14) ] = 112 236 + x2 + y2 โ€“ 224 = 112 x2 + y2 = 112 โ€“ 236 + 224 x2 + y2 = 100 From (1) x + y = 14 Squaring both sides (x + y)2 = 142 x2 + y2 + 2xy = 196 100 + 2xy = 196 2xy = 196 โ€“ 100 2xy = 96 xy = 1๏ทฎ2๏ทฏ ร— 96 xy = 48 x = 48๏ทฎ๐‘ฆ๏ทฏ Putting (3) in (1) x + y = 14 48๏ทฎ๐‘ฆ๏ทฏ + y = 14 48 + y2 = 14y y2 โ€“ 14y + 48 = 0 y2 โ€“ 6y โ€“ 8y + 48 = 0 y(y โ€“ 6) โ€“ 8(y โ€“ 6) = 0 (y โ€“ 6)(y โ€“ 8) = 0 So, y = 6 & y = 8 For y = 6 x = 48๏ทฎ๐‘ฆ๏ทฏ = 48๏ทฎ6๏ทฏ = 8 Hence x = 8, y = 6 are the remaining two observations For y = 8 x = 48๏ทฎ๐‘ฆ๏ทฏ = 48๏ทฎ8๏ทฏ = 6 Hence, x = 6, y = 8 are the remaining two observations Thus, remaining observations are 6 & 8

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.