Question

The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3, and 8 then, a ratio of other two variables is?
A. 4 : 9
B. 6 :7
C. 5 : 8
D. 10 : 3

Answer 1

According to given in the question we have to find the ratio of other two variables when the mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3, and 8 so, first of all we have to let the other numbers of observations.
Now, we have to find the mean of all the five observations with the help of the formula to find the mean of n observations as given below:

Formula used:
Mean$= \dfrac{{\sum\limits_{i = 1}^n {{X_i}} }}{n}.....................(a)$
Where n is the number of the observations as given in the question.
Now, As given in the question that the variance is 9.20 hence, we have to apply the formula to find the variance of given observations so that we can obtain both of the two required observations we let by placing the value of variance as given in the question.

Variance ${\sigma ^2} = \dfrac{{\sum\limits_{i = 1}^n {X_i^2} }}{n} - \dfrac{{\sum\limits_{i = 1}^n {{X_i}} }}{n}..............(b)$
Now, on solving the obtained expressions we can determine both of the required observations to find the required ratio.

Complete step by step solution:
Given,
Mean of the five observation is = 5
Variance is = 9.20
Step 1: First of all we have to let that the other two observations are ${x_1}$and ${x_2}$as explained in the solution hint.
Step 2: Now, we have to find the mean of all the five observations with the help of the formula (a) as mentioned in the solution hint.
Mean $\Rightarrow \dfrac{{\sum\limits_{i = 1}^{n = 5} {{X_i}} }}{5} = 5$
On solving the expression obtained just above,
$\Rightarrow 1 + 3 + 8 + {x_1} + {x_2} = 5 \times 5 \\\ \Rightarrow {x_1} + {x_2} = 25 - 12 \\\ \Rightarrow {x_1} + {x_2} = 13............(1) \\\$
Step 2: Now, to find the value of both of the two other terms we have to use the variance which is 9.02 as given in the question with the help of the formula (b) as mentioned in the solution hint.
$\Rightarrow \dfrac{{\sum\limits_{i = 1}^{n = 5} {X_i^2} }}{5} - 25 = 9.02$
Step 3: On solving the expression obtained just above with the help of find L.C.M
$\Rightarrow \sum\limits_{}^{} {X_i^2} - 125 = 9.02 \times 5 \\\ \Rightarrow \sum\limits_{}^{} {X_i^2} = 171 \\\$
Now, Now, we have to substitute the values ${X_i}$in the expression as obtained just above,
$\Rightarrow x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 = 171$
On substituting the values in the expression as obtained just above,
$\Rightarrow x_1^2 + x_2^2 + {1^2} + {3^2} + {8^2} = 171 \\\ \Rightarrow x_1^2 + x_2^2 + 1 + 9 + 64 = 171 \\\ \Rightarrow x_1^2 + x_2^2 = 171 - 74 \\\ \Rightarrow x_1^2 + x_2^2 = 97.............(2) \\\$
Step 4: Now, to find the values of ${x_1}$and ${x_2}$which are the other two terms we have to obtain so, we have to solve the obtained expressions (1) and (2) hence,
As we know that,
$\Rightarrow {({x_1} + {x_2})^2} - 2{x_1}{x_2} = {x^2}_1 + {x^2}_2$…………………(3)
Step 5: On substituting all the values in expression (3) as obtained in the step 4.
$\Rightarrow {(13)^2} - 2{x_1}{x_2} = 97 \\\ \Rightarrow 169 - 2{x_1}{x_2} = 97 \\\ \Rightarrow - 2{x_1}{x_2} = 97 - 169 \\\ \Rightarrow {x_1}{x_2} = \dfrac{{72}}{2} \\\ \Rightarrow {x_1}{x_2} = 36 \\\$
Step 6: on multiplying with ${x_2}$in the expression 1 we can obtain the value of ${x_2}$hence,
$\Rightarrow {x_2}({x_1} + {x_2}) = 13{x_2} \\\ \Rightarrow {x_1}{x_2} + x_2^2 = 13{x_2} \\\$
On substituting the value of ${x_1}{x_2}$as obtained in the step 5 in the expression obtained just above,
$\Rightarrow 36 + x_2^2 = 13{x_2} \\\ \Rightarrow x_2^2 - 13{x_2} + 36 = 0...............(4) \\\$
Step 7: On solving the quadratic expression (4) obtained in step 6.
$\Rightarrow x_2^2 - (9 + 4){x_2} + 36 = 0 \\\ \Rightarrow x_2^2 - 9{x_2} - 4{x_2} + 36 = 0 \\\ \Rightarrow {x_2}({x_2} - 9) - 4({x_2} - 9) = 0 \\\ \Rightarrow ({x_2} - 4)({x_2} - 9) = 0 \\\$
Hence, on solving the roots obtained are,
$\Rightarrow ({x_2} - 9) = 0 \\\ \Rightarrow {x_2} = 9 \\\$
And the another root is,
$\Rightarrow ({x_2} - 4) = 0 \\\ \Rightarrow {x_2} = 4 \\\$
Hence,
Required ratio of ${x_1}$and ${x_2}$is $\dfrac{{{x_1}}}{{{x_2}}} = \dfrac{4}{9}$
Final solution: Hence, with the help of the formula (a) and formula (b) we have obtained the value of required ratio $\dfrac{{{x_1}}}{{{x_2}}} = \dfrac{4}{9}$

So, the correct answer is “Option A”.

Note: 1. Variance is a measurement of the spread between a given number of the data set, that is, it measures how far each number in the set is from the mean and hence, from every other number in the set.
2. The mean is the average of the data set and the mode is the most common number in that data set mean can be obtained by dividing the sum of all the given data by the total number of data.