Last updated at Dec. 12, 2016 by Teachoo

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Misc 1 The mean and variance of eight observations are 9 and 9.25, respectively. If six of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations. Let the other two observations be x and y. Therefore, our observations are 6, 7, 10, 12, 12 , 13, x, y. Given Mean = 9 i.e. ๐๐ข๐ ๐๐ ๐๐๐ ๐๐๐ฃ๐๐ก๐๐๐๐ ๏ทฎ๐๐ข๐๐๐๐ ๐๐ ๐๐๐ ๐๐๐ฃ๐๐ก๐๐๐๐ ๏ทฏ = 9 6 + 7 + 10 + 12 + 12 + 13 + ๐ฅ + ๐ฆ๏ทฎ8๏ทฏ = 9 60 + x + y = 9 ร 8 x + y = 72 โ 60 x + y = 12 Also, Given Variance = 9.25 1๏ทฎn๏ทฏ ๏ทฎ๏ทฎ( ๐ฅ๏ทฎ๐๏ทฏ๏ทฏโ ๐ฅ๏ทฏ)๏ทฎ2๏ทฏ = 9.25 1๏ทฎ8๏ทฏ ๏ทฎ๏ทฎ( ๐ฅ๏ทฎ๐๏ทฏ๏ทฏโ9)๏ทฎ2๏ทฏ = 9.25 1๏ทฎ8๏ทฏ [(6โ9)2 +(7โ9)2+(10โ9)2+(12โ9)2+(12โ9)2+(13โ9)2+(xโ9)2+(yโ9)2]=9.25 1๏ทฎ8๏ทฏ [ (โ3)2 + (โ2)2 + (1)2 + (3)2 + (3)2 + (4)2 + (x โ 9)2 + (y โ 9)2 ] = 9.25 1๏ทฎ8๏ทฏ [9 + 4 + 1 + 9 + 9 + 16 + x2 + (9)2 - 2(9)x + y2 + (9)2 - 2(9)y] = 9.25 [ 48 + x2 + 81 โ 18x + y2 + 81 โ 18y] = 9.25 ร 8 [ 210 + x2 + y2 โ 18y โ 18x ] = 74 [ 210 + x2 + y2 โ 18(x + y) ] = 74 [ 210 + x2 + y2 โ 18(12) ] = 74 210 + x2 + y2 โ 216 = 74 x2 + y2 = 74 โ 210 + 216 x2 + y2 = 80 From (1) x + y = 12 Squaring both sides (x + y)2 = 122 x2 + y2 + 2xy = 144 80 + 2xy = 144 2xy = 144 โ 80 2xy = 64 xy = 1๏ทฎ2๏ทฏ ร 64 xy = 32 x = 32๏ทฎ๐ฆ๏ทฏ Putting (3) in (1) x + y = 12 32๏ทฎ๐ฆ๏ทฏ + y = 12 32 + y2 = 12y y2 โ 12y + 32 = 0 y2 โ 8y โ 4y + 32 = 0 y(y โ 8) โ 4(y โ 8) = 0 (y โ 4)(y โ 8) = 0 So, y = 4 & y = 8 For y = 4 x = 32๏ทฎ๐ฆ๏ทฏ = 32๏ทฎ4๏ทฏ = 8 Hence x = 8, y = 4 are the remaining two observations For y = 8 x = 32๏ทฎ๐ฆ๏ทฏ = 32๏ทฎ8๏ทฏ = 4 Hence, x = 4, y = 8 are the remaining two observations Thus, remaining observations are 4 & 8

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.