© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Write down the given data
The six observations given are 7, 10, 11, 15, a, and b. We know:
Mean of these observations = 10
Variance of these observations = $ \tfrac{20}{3} $
Step 2: Express the condition for the mean
The mean of the six observations is given by:
$ 10 \;=\; \frac{7 + 10 + 11 + 15 + a + b}{6} $
Multiply both sides by 6:
$ 60 \;=\; 7 + 10 + 11 + 15 + a + b $
Adding up the constants (7+10+11+15 = 43):
$ 60 \;=\; 43 + a + b $
Hence,
$ a + b = 17 \quad \dots (i)
Step 3: Express the condition for the variance
The formula for the variance of n observations $ x_1, x_2, \dots, x_n $ is:
$ \text{Variance} = \frac{x_1^2 + x_2^2 + \dots + x_n^2}{n} - \left(\frac{x_1 + x_2 + \dots + x_n}{n}\right)^2
$
In our problem, n = 6. So:
$ \frac{20}{3} \;=\; \frac{7^2 + 10^2 + 11^2 + 15^2 + a^2 + b^2}{6} \;-\; (10)^2
$
Compute the squares of the known numbers: $7^2 = 49, 10^2 = 100, 11^2 = 121, 15^2 = 225.$ Summing these:
$ 49 + 100 + 121 + 225 = 495
So the equation becomes:
$ \frac{20}{3} \;=\; \frac{495 + a^2 + b^2}{6} \;-\; 100
$
Step 4: Simplify to find $a^2 + b^2$
First, rearrange and multiply by 6 to clear the fraction:
$ 6 \times \frac{20}{3} = 6 \times \left(\frac{495 + a^2 + b^2}{6} - 100\right)
$
$ 40 = 495 + a^2 + b^2 - 600
$
$ 40 = a^2 + b^2 - 105
$
$ a^2 + b^2 = 145 \quad \dots (ii)
Step 5: Solve the system of equations
We have two equations from steps (i) and (ii):
$ a + b = 17 $
$ a^2 + b^2 = 145 $
We can solve this system by noting that:
$ (a + b)^2 = a^2 + 2ab + b^2
$
From (i), $ a + b = 17 \implies (a + b)^2 = 17^2 = 289. $
So,
$ 289 = a^2 + 2ab + b^2. $
But from (ii), $ a^2 + b^2 = 145. $ Subtracting (ii) from the above:
$ 289 - 145 = 2ab \implies 144 = 2ab \implies ab = 72.
Now we have:
$ a + b = 17 $
$ ab = 72 $
This system can be factored or solved by trial (for integer solutions) or by forming a quadratic. For example:
$ a(17 - a) = 72 \implies 17a - a^2 = 72 \implies a^2 - 17a + 72 = 0.
This quadratic factors as $(a - 8)(a - 9) = 0$, giving $a = 8$ or $a = 9.$ Correspondingly, $b = 9$ or $b = 8.$
Step 6: Find $|a - b|$
Either $ (a,b) = (9,8) $ or $ (8,9). $ In both cases,
$ |\,a - b\,| = |\,9 - 8\,| = 1.
Hence, the value of $|\,a - b\,|$ is 1.
Final Answer
The correct value of $|\,a - b\,|$ is 1.
