Indirect questions - finding remaining observations
Indirect questions - finding remaining observations
Last updated at May 8, 2024 by Teachoo
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. ððĒð ðð ððð ðððĢððĄðððð ï·ŪððĒðððð ðð ððð ðððĢððĄðððð ï·Ŋ = 8 2 + 4 + 10 + 12 + 14 + ðĨ + ðĶï·Ū7ï·Ŋ = 8 42 + x + y = 7 Ã 8 x + y = 56 â 42 x + y = 14 Also, Given Variance = 16 1ï·Ūnï·Ŋ ï·Ūï·Ū( ðĨï·Ūðï·Ŋï·Ŋâ ðĨï·Ŋ)ï·Ū2ï·Ŋ = 16 1ï·Ū7ï·Ŋ ï·Ūï·Ū( ðĨï·Ūðï·Ŋï·Ŋâ8)ï·Ū2ï·Ŋ = 16 1ï·Ū7ï·Ŋ [(2 â 8)2 +(4 â 8)2+(10 â 8)2+(12 â 8)2 +(14 â 8)2+(x â 8)2 +(y â 8)2] = 16 1ï·Ū7ï·Ŋ [ (â6)2 + (â4)2 + (2)2 + (4)2 + (6)2 + (x â 8)2 + (y â 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 â 16x + y2 + 64 â 16y] = 16 Ã 7 [ 236 + x2 + y2 â 16y â 16x ] = 112 [ 236 + x2 + y2 â 16(x + y) ] = 112 [ 236 + x2 + y2 â 16(14) ] = 112 236 + x2 + y2 â 224 = 112 x2 + y2 = 112 â 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 â 100 2xy = 96 xy = 1ï·Ū2ï·Ŋ Ã 96 xy = 48 x = 48ï·ŪðĶï·Ŋ Putting (3) in (1) x + y = 14 48ï·ŪðĶï·Ŋ + y = 14 48 + y2 = 14y y2 â 14y + 48 = 0 y2 â 6y â 8y + 48 = 0 y(y â 6) â 8(y â 6) = 0 (y â 6)(y â 8) = 0 So, y = 6 & y = 8 For y = 6 x = 48ï·ŪðĶï·Ŋ = 48ï·Ū6ï·Ŋ = 8 Hence x = 8, y = 6 are the remaining two observations For y = 8 x = 48ï·ŪðĶï·Ŋ = 48ï·Ū8ï·Ŋ = 6 Hence, x = 6, y = 8 are the remaining two observations Thus, remaining observations are 6 & 8