Misc 1 - Mean and variance of eight observations are 9, 9.25 - Indirect questions - finding remaining observations

Slide2.JPG
Slide3.JPG Slide4.JPG Slide5.JPG

  1. Chapter 15 Class 11 Statistics
  2. Serial order wise
Ask Download

Transcript

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. 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 = 9 6 + 7 + 10 + 12 + 12 + 13 + ๐‘ฅ + ๐‘ฆ๏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( ๐‘ฅ๏ทฎ๐‘–๏ทฏ๏ทฏโˆ’ ๐‘ฅ๏ทฏ)๏ทฎ2๏ทฏ = 9.25 1๏ทฎ8๏ทฏ ๏ทฎ๏ทฎ( ๐‘ฅ๏ทฎ๐‘–๏ทฏ๏ทฏโˆ’9)๏ทฎ2๏ทฏ = 9.25 1๏ทฎ8๏ทฏ [(6โ€“9)2 +(7โ€“9)2+(10โ€“9)2+(12โ€“9)2+(12โ€“9)2+(13โ€“9)2+(xโ€“9)2+(yโ€“9)2]=9.25 1๏ทฎ8๏ทฏ [ (โ€“3)2 + (โ€“2)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๏ทฎ๐‘ฆ๏ทฏ Putting (3) in (1) x + y = 12 32๏ทฎ๐‘ฆ๏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๏ทฎ๐‘ฆ๏ทฏ = 32๏ทฎ4๏ทฏ = 8 Hence x = 8, y = 4 are the remaining two observations For y = 8 x = 32๏ทฎ๐‘ฆ๏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's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.
Jail