The mean and standard deviation of 15 observations were found to be 12 and 3 respectively. On rechecking it was found that an observation was read as 10 in place of 12 . If μ and σ 2 denote the mean and variance of the correct observations respectively, then 15 μ + μ 2 + σ 2 is equal to _________.

Step-by-step solution

Given: Incorrect mean, x = 12 and incorrect standard deviation, σ ' = 3 . ⇒ Σx i 15 = 12 ⇒ Σx i = 180 Correct mean is given by, μ = Σx i - 10 + 12 15 ⇒ μ = 182 15 Also, σ ' = ∑ x i 2 15 - 144 = 3 ⇒ ∑ x i 2 15 = 9 + 144 ⇒ ∑ x i 2 = 2295 Correct Σx i 2 = 2295 - 100 + 144 = 2339 σ 2 (correct variance) = 2339 15 - 182 × 182 15 × 15 ⇒ 15 μ + μ 2 + σ 2 = 15 182 15 + 182 × 182 15 × 15 + 2339 15 - 182 × 182 15 × 15 ⇒ 15 μ + μ 2 + σ 2 = 15 182 15 + 2339 15 ⇒ 15 μ + μ 2 + σ 2 = 2521