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

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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. 𝑆𝑢𝑚 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠﷮𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠﷯ = 4.4 1 + 2 + 6 + 𝑥 + 𝑦﷮5﷯ = 4.4 9 + x + y = 4.4 × 5 x + y = 22 – 9 x + y = 13 Also, Given Variance = 8.24 1﷮n﷯ ﷮﷮( 𝑥﷮𝑖﷯﷯− 𝑥﷯)﷮2﷯ = 8.24 1﷮5﷯ ﷮﷮( 𝑥﷮𝑖﷯﷯−4.4)﷮2﷯ = 8.24 1﷮5﷯ [ (1 – 4.4)2 + (2 – 4.4)2 + (6 – 4.4)2 + (x – 4.4)2 + (y – 4.4)2 ] = 8.24 1﷮5﷯ [ (–3.4)2 + (–2.4)2 + (1.6)2 + (x – 4.4)2 + (y – 4.4)2 ] = 8.24 1﷮5﷯ [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 = 1﷮2﷯ × 72 xy = 36 x = 36﷮𝑦﷯ Putting (3) in (1) x + y = 13 36﷮𝑦﷯ + 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 = 36﷮𝑦﷯ = 36﷮4﷯ = 9 Hence x = 9, y = 4 are the remaining two observations For y = 9 x = 36﷮𝑦﷯ = 36﷮9﷯ = 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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.