Question

The mean of five observations is 4, and their variance is 5.2. If three of them are 1,2,6, then the other two are:
1) 2 and 9
2) 3 and 8
3)4 and 7
4) 5 and 6

Answer 1

Hint: This question is a direct question from the chapter statistics. The concepts like mean and variance are applied. Mean is defined as the average of the discrete values given, while variance is the average of the squared differences from the mean. The formulas used in this question are:
Mean, \[\bar x = \dfrac{{{x_1} + {x_2} + ... + {x_n}}}{n}\]
Variance \[{\sigma ^2} = \dfrac{{\sum {x_i^2} }}{n} - {(\bar x)^2}\] \[\sigma \] is defined as the standard deviation

Complete step-by-step answer:
Now, let us begin the question by assuming the two numbers to be and.
Now, the mean is given by
\[ \Rightarrow \bar x = \dfrac{{{x_1} + {x_2} + ... + {x_n}}}{n}\]
By putting the values know to us, we get,
\[ \Rightarrow 4 = \dfrac{{1 + 2 + 6 + x + y}}{5}\]
By cross multiplying both the sides, we get,
\[ \Rightarrow 20 = 9 + x + y\]
By taking the constant term on the left-hand side, we get,
\[ \Rightarrow x + y = 11\]
Now, taking x on the other side of the equation we get,
\[ \Rightarrow y = 11 - x - - - - (i)\]
Now, the variance is given by
\[ \Rightarrow {\sigma ^2} = \dfrac{{\sum {x_i^2} }}{n} - {(\bar x)^2}\]
By putting the values known to us, we get,
\[ \Rightarrow 5.2 = \dfrac{{{1^2} + {2^2} + {6^2} + {x^2} + {y^2}}}{5} - {(4)^2}\]
Now, solving the squares on the right-hand side
\[ \Rightarrow 5.2 = \dfrac{{41 + {x^2} + {y^2}}}{5} - 16\]
By taking five as the denominator on the right-hand side, we get,
\[ \Rightarrow 5.2 = \dfrac{{41 + {x^2} + {y^2} - 80}}{5}\]
By cross multiplying five on both the sides we get,
\[ \Rightarrow 26 = {x^2} + {y^2} - 39\]
Now, taking the constants on one side of the equation
\[ \Rightarrow {x^2} + {y^2} = 65\]
By replacing the value of y form equation we get,
\[ \Rightarrow {x^2} + {(11 - x)^2} = 65\]
Simplifying the equation we get,
\[ \Rightarrow {x^2} + {x^2} - 22x + 121 = 65\]
\[ \Rightarrow 2{x^2} - 22x + 56 = 0\]
Dividing by two on both sides,
\[ \Rightarrow {x^2} - 11x + 28 = 0\]
\[ \Rightarrow {x^2} - 7x - 4x + 28 = 0\]
By factoring we get,
\[ \Rightarrow x(x - 7) - 4(x - 7) = 0\]
\[ \Rightarrow (x - 4)(x - 7) = 0\]
Thus, the value of x comes out to be 4 or 7
Therefore, the value of y will be 7 or 4.
Therefore, option(03) is the correct answer.
\[ \Rightarrow x = 4,7\]
So, the correct answer is “Option 3”.

Note: This question involves the basic concepts of statistics like mean and variance. One should be well versed with these concepts before solving the question. Care should be taken while shifting the terms in the equation and splitting the middle term to factorise the equation.