Question

The mean and the variance of five observations are 4 and 5.20, respectively. If three of the observations are 3, 4 and 4 ; then the absolute value of the difference of the other two observations, is :

• A. 1
• B. 7
• C. 3
• D. 5

Hint:

Use \(\sum x=20\) and \(\sum x^2=5(\sigma^2+\bar x^2)\) to get the sum and sum of squares of the two unknown values.

Answer 1

The Correct Option is B

Let the two unknown observations be a and b. Mean 4 gives \(3+4+4+a+b=20\Rightarrow a+b=9\).
Variance: \(\frac{\sum x^2}{5}-16=5.2\Rightarrow\sum x^2=106\), so \(9+16+16+a^2+b^2=106\Rightarrow a^2+b^2=65\).
\(2ab=(a+b)^2-(a^2+b^2)=81-65=16\). \((a-b)^2=a^2+b^2-2ab=65-16=49\).
\(|a-b|=7\) (the numbers are 1 and 8).
Answer: (2) 7