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. 𝑆ð‘Ē𝑚 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟ð‘Ģð‘Žð‘Ąð‘–ð‘œð‘›ð‘ ï·Ū𝑁ð‘Ē𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟ð‘Ģð‘Žð‘Ąð‘–ð‘œð‘›ð‘ ï·Ŋ = 8 2 + 4 + 10 + 12 + 14 + ð‘Ĩ + ð‘Ķï·Ū7ï·Ŋ = 8 42 + x + y = 7 × 8 x + y = 56 – 42 x + y = 14 Also, Given Variance = 16 1ï·Ūnï·Ŋ ï·Ūï·Ū( ð‘Ĩï·Ū𝑖ï·Ŋï·Ŋ− ð‘Ĩï·Ŋ)ï·Ū2ï·Ŋ = 16 1ï·Ū7ï·Ŋ ï·Ūï·Ū( ð‘Ĩï·Ū𝑖ï·Ŋï·Ŋ−8)ï·Ū2ï·Ŋ = 16 1ï·Ū7ï·Ŋ [(2 – 8)2 +(4 – 8)2+(10 – 8)2+(12 – 8)2 +(14 – 8)2+(x – 8)2 +(y – 8)2] = 16 1ï·Ū7ï·Ŋ [ (–6)2 + (–4)2 + (2)2 + (4)2 + (6)2 + (x – 8)2 + (y – 8)2 ] = 16 1ï·Ū7ï·Ŋ [36 + 16 + 4 + 16 + 36 + x2 + (8)2 - 2(8)x + y2 + (8)2 - 2(8)y] = 16 [ 108 + x2 + 64 – 16x + y2 + 64 – 16y] = 16 × 7 [ 236 + x2 + y2 – 16y – 16x ] = 112 [ 236 + x2 + y2 – 16(x + y) ] = 112 [ 236 + x2 + y2 – 16(14) ] = 112 236 + x2 + y2 – 224 = 112 x2 + y2 = 112 – 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 – 100 2xy = 96 xy = 1ï·Ū2ï·Ŋ × 96 xy = 48 x = 48ï·Ūð‘Ķï·Ŋ Putting (3) in (1) x + y = 14 48ï·Ūð‘Ķï·Ŋ + y = 14 48 + y2 = 14y y2 – 14y + 48 = 0 y2 – 6y – 8y + 48 = 0 y(y – 6) – 8(y – 6) = 0 (y – 6)(y – 8) = 0 So, y = 6 & y = 8 For y = 6 x = 48ï·Ūð‘Ķï·Ŋ = 48ï·Ū6ï·Ŋ = 8 Hence x = 8, y = 6 are the remaining two observations For y = 8 x = 48ï·Ūð‘Ķï·Ŋ = 48ï·Ū8ï·Ŋ = 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.