Example 17 - Mean of 5 observations is 4.4, variance is 8.24 - Indirect questions - finding remaining observations

Example 17 - Chapter 15 Class 11 Statistics - Part 2
Example 17 - Chapter 15 Class 11 Statistics - Part 3
Example 17 - Chapter 15 Class 11 Statistics - Part 4
Example 17 - Chapter 15 Class 11 Statistics - Part 5


Transcript

Example 14 The mean of 5 observations is 4.4 and their variance is 8.24. If three of the observations are 1, 2 and 6, find the other two observations. Let the other two observations be x and y. Therefore, our observations are 1, 2, 6, x, y. Given Mean = 4.4 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 = 4.4 1 + 2 + 6 + ð‘Ĩ + ð‘Ķï·Ū5ï·Ŋ = 4.4 9 + x + y = 4.4 × 5 x + y = 22 – 9 x + y = 13 Also, Given Variance = 8.24 1ï·Ūnï·Ŋ ï·Ūï·Ū( ð‘Ĩï·Ū𝑖ï·Ŋï·Ŋ− ð‘Ĩï·Ŋ)ï·Ū2ï·Ŋ = 8.24 1ï·Ū5ï·Ŋ ï·Ūï·Ū( ð‘Ĩï·Ū𝑖ï·Ŋï·Ŋ−4.4)ï·Ū2ï·Ŋ = 8.24 1ï·Ū5ï·Ŋ [ (1 – 4.4)2 + (2 – 4.4)2 + (6 – 4.4)2 + (x – 4.4)2 + (y – 4.4)2 ] = 8.24 1ï·Ū5ï·Ŋ [ (–3.4)2 + (–2.4)2 + (1.6)2 + (x – 4.4)2 + (y – 4.4)2 ] = 8.24 1ï·Ū5ï·Ŋ [11.56 + 5.76 + 2.56 + x2 + (4.4)2 - 2(4.4)x + y2 + (4.4)2 - 2(4.4)y] = 8.24 [ 19.88 + x2 + 19.36 – 8.8x + y2 + 19.36 – 8.8y] = 8.24 × 5 [ 19.88 + 19.36 + 19.36 + x2 + y2 – 8.8y – 8.8x ] = 41.2 [ 58.6 + x2 + y2 – 8.8(x + y) ] = 41.2 [ 58.6 + x2 + y2 – 8.8(13) ] = 41.2 58.6 + x2 + y2 – 114.4 = 41.2 x2 + y2 = 114.4 + 41.2 – 58.6 x2 + y2 = 97 From (1) x + y = 13 Squaring both sides (x + y)2 = 132 x2 + y2 + 2xy = 169 97 + 2xy = 169 2xy = 169 – 97 2xy = 72 xy = 1ï·Ū2ï·Ŋ × 72 xy = 36 x = 36ï·Ūð‘Ķï·Ŋ Putting (3) in (1) x + y = 13 36ï·Ūð‘Ķï·Ŋ + y = 13 36 + y(y) = 13(y) 36 + y2 = 13y y2 – 13y + 36 = 0 y2 – 9y – 4y + 36 = 0 y(y – 9) – 4(y – 9) = 0 (y – 4)(y – 9) = 0 So, y = 4 & y = 9 For y = 4 x = 36ï·Ūð‘Ķï·Ŋ = 36ï·Ū4ï·Ŋ = 9 Hence x = 9, y = 4 are the remaining two observations For y = 9 x = 36ï·Ūð‘Ķï·Ŋ = 36ï·Ū9ï·Ŋ = 4 Hence, x = 4, y = 9 are the remaining two observations Thus, remaining observations are 4 & 9

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.