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.
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Step-by-Step Solution
Step 1: Set up equations for mean and variance
We are given the mean and variance of 5 observations. Let the two unknown observations be x and y. We can set up two equations based on the definitions of mean and variance. The mean is the sum of all observations divided by the number of observations, and the variance is the mean of the squares of the observations minus the square of the mean.
Step 2: Simplify the mean equation
First, let's simplify the equation for the mean. Multiply both sides by 5 and sum the known observations. This gives us a linear equation relating x and y.
Step 3: Simplify the variance equation
Next, simplify the equation for the variance. Calculate the squares of the known observations and the square of the mean. Rearrange the terms to get a quadratic equation involving x2 and y2.
Step 4: Solve the system of equations
Now we have a system of two equations: one linear and one quadratic. Substitute the expression for x from equation (1) into equation (2). This will result in a quadratic equation in terms of y.
Step 5: Solve the quadratic equation for y
Solve the quadratic equation y2−13y+36=0 by factoring. We look for two numbers that multiply to 36 and add up to -13. These numbers are -4 and -9, so the factors are (y−4) and (y−9). This gives us two possible values for y.
Step 6: Find the corresponding values for x
Substitute the values of y back into equation (1) (x=13−y) to find the corresponding values for x. This shows that the two unknown observations are 4 and 9, regardless of which variable we assign them to.