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Indirect questions - Multiplication of observation
Indirect questions - Multiplication of observation
Last updated at May 29, 2023 by Teachoo
Misc 3 The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations. Let the observations be 𝑥1, 𝑥2, 𝑥3, ..., 𝑥6 and 𝑥 be their mean. Given that Mean = 𝑥 = 8, Standard deviation = 4 If each observation is multiplied by 3 , we get new observations, Let the new observations be 𝑦1, 𝑦2, 𝑦3, ..., 𝑦6 where 𝑦𝑖 = 3( 𝑥𝑖) Calculating new mean New mean = 1𝑛 𝑦𝑖 𝑦 = 16 3𝑥𝑖 𝑦 = 3 × 16 𝑥𝑖 𝑦 = 3 𝑥 𝑦 = 3 × 8 𝑦 = 24 So, New Mean = 24 Calculating new standard deviation First we find variance of the new observations i.e. New Variance = 1n ( 𝑦𝑖− 𝑦)2 Given Old Standard deviation = 4 So, Old Variance = 42 = 16 Now, Old Variance = 1𝑛 ( 𝑥𝑖− 𝑥)2 16 = 16 ( 𝑥𝑖− 𝑥)2 16 × 6 = ( 𝑥𝑖− 𝑥)2 96 = ( 𝑥𝑖− 𝑥)2 ( 𝑥𝑖− 𝑥)2 = 96 ( 13 𝑦𝑖− 13 𝑦)2 = 96 ( 13 (𝑦𝑖− 𝑦))2 = 96 132 ( 𝑦𝑖− 𝑦)2 = 96 19 ( 𝑦𝑖− 𝑦)2 = 96 ( 𝑦𝑖− 𝑦)2 = 96 × 9 ( 𝑦𝑖− 𝑦)2 = 864 So, New Variance = 1𝑛 ( 𝑦𝑖− 𝑦)2 = 16 × 864 = 144 Hence, New standard deviation = 𝑁𝑒𝑤 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 144 = 12