Question

The mean and variance of seven observations are 8 and 16, respectively. If 5 of the observations are $2,4,10,12,14$ then, the product of the remaining two observations.

Answer 1

Hint: According to the question we have to determine the product of the remaining two observations. So, first of all we have to let the two other observations.
Now, we have to find the mean of all the observations with the help of the formula as given below:

Formula used: Mean $\overline x = \dfrac{{\sum\limits_{i = 1}^{i = n} {{x_i}} }}{n}.................(a)$
Hence, with the help of the formula above, can obtain the required mean for the given data.
Now, we have to find the variance with the help of the formula to find the variance as given below:
Formula used:
Variance$ = \dfrac{{\sum\limits_{i = 1}^{i = n} {{{({x_i} - \overline x )}^2}} }}{n}...................(b)$
As we know that variance is already given in the question hence, with the help of the formula to find the variance we can obtain a quadratic expression in terms of the two observations we let.
Now, on solving the both of the expressions obtained we can find the both of the observations we let and also their product.

Complete step-by-step solution:
Given,
5 of the observations are $2,4,10,12,14$
Mean = 8
Variance = 16
Step 1: First of all we have to let the remaining two observations be x and y.
Step 2: Now, we have to find the mean of all the seven observations which are $2,4,10,12,14,x,y$ with the help of the formula (a) to find the mean of the given data as mentioned in the solution hint.
$\overline x = \dfrac{{2 + 4 + 10 + 12 + 14 + x + y}}{7}$
On, substituting the value of $\overline x $ which is the mean as given in the question,
$
   \Rightarrow 8 = \dfrac{{2 + 4 + 10 + 12 + 14 + x + y}}{7} \\
   \Rightarrow 8 = \dfrac{{42 + x + y}}{7}
 $
On applying cross-multiplication in the expression obtained just above,
$
   \Rightarrow x + y = 56 - 42 \\
   \Rightarrow x + y = 14......................(1)
 $
Step 3: Now, we have to find the value of ${({x_i} - \overline x )^2}$ which is obtained below:

${x_i}$	$({x_i} - \overline x )$
2	$2 - 8 = - 6$
4	$4 - 8 = - 4$
10	$10 - 8 = 2$
12	$12 - 8 = 4$
14	$14 - 8 = 6$
X	$x - 8$
y	$y - 8$

Step 4: Now, we have to find the variance of the given data with the help of the formula (b) as mentioned in the solution hint and as we know that the variance is 16 hence on substituting all the values.
$ \Rightarrow 16 = \dfrac{{\sum\limits_{i = 1}^{i = n} {\left[ {{{( - 6)}^2} + {{( - 4)}^2} + {{(2)}^2} + {{(4)}^2} + {{(6)}^2} + {x^2} + {y^2} - 2 \times 8(x + y) + 2 \times {{(8)}^2}} \right]} }}{7}$
Step 5: Now, on substituting the values of $(x + y)$from the step 2 in the expression obtained in the step 4,
$
   \Rightarrow 16 = \dfrac{{\left[ {{{( - 6)}^2} + {{( - 4)}^2} + {{(2)}^2} + {{(4)}^2} + {{(6)}^2} + {x^2} + {y^2} - 2 \times 8(14) + 2 \times {{(8)}^2}} \right]}}{7} \\
   \Rightarrow 16 = \dfrac{{\left[ {36 + 16 + 4 + 16 + 36 + {x^2} + {y^2} - 224 + 2 \times 64} \right]}}{7} \\
   \Rightarrow 16 = \dfrac{{\left[ {108 + {x^2} + {y^2} - 224 + 128} \right]}}{7} \\
   \Rightarrow 16 = \dfrac{{12 + {x^2} + {y^2}}}{7}
 $
Applying cross-multiplication in the expression obtained just above,
$
   \Rightarrow {x^2} + {y^2} = 112 - 12 \\
   \Rightarrow {x^2} + {y^2} = 100...............(2)
 $
Now, from the expression (1), we obtain
$ \Rightarrow {x^2} + {y^2} + 2xy = 196................(3)$
Step 6: Now, on subtracting equation (2) from equation (3) we can obtain the required multiplication of remaining two observations.
$
   \Rightarrow ({x^2} + {y^2} + 2xy) - ({x^2} + {y^2}) = 196 - 100 \\
   \Rightarrow 2xy = 96 \\
   \Rightarrow xy = \dfrac{{96}}{2} \\
   \Rightarrow xy = 48
 $

Hence, with the help of the formula (a) and (b) we can obtain the product of the remaining two observations $xy = 48$

Note: 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.
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.