We need to find the new mean a and variance b of the remaining 6 observations, and then calculate a + b.
Step 1: Calculate the Sum of the 7 Observations
The mean of 7 observations is given by: μ = sum of observations / 7 Given that the mean is 8: 8 = sum of observations / 7 Thus, the sum of the observations is: sum of observations = 8 × 7 = 56
Step 2: Calculate the Sum of Squared Deviations for the 7 Observations
The formula for variance is: σ² = Σ(xᵢ - μ)² / 7 We are told that the variance is 16, so: 16 = Σ(xᵢ - 8)² / 7 Multiplying both sides by 7: Σ(xᵢ - 8)² = 16 × 7 = 112 This represents the sum of squared deviations of the 7 observations from the mean.
Step 3: Omit the Observation 14
When the number 14 is omitted, we are left with 6 observations. We need to find the new sum of squared deviations and the new mean for these 6 observations.
New Mean (a):
After removing the observation 14, the new sum of the remaining 6 observations is: new sum of observations = 56 - 14 = 42 The new mean of the remaining 6 observations is: a = new sum of observations / 6 = 42 / 6 = 7
New Variance (b):
To calculate the new variance, we first subtract the squared deviation of 14 from the total sum of squared deviations. The squared deviation of 14 from the mean is: (14 - 8)² = 6² = 36 So, the new sum of squared deviations for the remaining 6 observations is: Σ(xᵢ - 8)² = 112 - 36 = 76
Now, the new variance b is: b = 76 / 6 = 38 / 3 ≈ 12.67
Step 4: Calculate a + b
Now, we calculate a + b, where a = 7 and b = 38 / 3. a + b = 7 + 38 / 3 = 21 / 3 + 38 / 3 = 59 / 3 This simplifies to approximately: a + b ≈ 19.67
