To solve this problem, we need to find the other two observations given the mean and variance of the five observations.
Given:
- The mean of 5 observations is 4.8.
- The variance of the 5 observations is 6.56.
- Three of the five observations are 1, 3, and 8.
Let's denote the other two observations as x and y.
Step 1: Calculate the sum of the five observations.
Mean = (Sum of the observations) / 5
4.8 = (1 + 3 + 8 + x + y) / 5
Sum of the observations = 4.8 × 5 = 24
Step 2: Set up an equation to find the sum of the other two observations.
Sum of the observations = 1 + 3 + 8 + x + y
24 = 12 + x + y
x + y = 12
Step 3: Calculate the variance of the five observations.
Variance = Σ(x - mean)^2 / (n - 1)
6.56 = [(1 - 4.8)^2 + (3 - 4.8)^2 + (8 - 4.8)^2 + (x - 4.8)^2 + (y - 4.8)^2] / 4
6.56 = [(-3.8)^2 + (-1.8)^2 + 3.2^2 + (x - 4.8)^2 + (y - 4.8)^2] / 4
6.56 = [14.44 + 3.24 + 10.24 + (x - 4.8)^2 + (y - 4.8)^2] / 4
26.24 = 14.44 + 3.24 + 10.24 + (x - 4.8)^2 + (y - 4.8)^2
(x - 4.8)^2 + (y - 4.8)^2 = 26.24 - 14.44 - 3.24 - 10.24
(x - 4.8)^2 + (y - 4.8)^2 = -1.68
Step 4: Solve the system of equations to find the values of x and y.
x + y = 12 (from Step 2)
(x - 4.8)^2 + (y - 4.8)^2 = -1.68 (from Step 3)
Solving this system of equations, we get:
x = 5
y = 7
Therefore, the other two observations are 5 and 7.
