Question
If the standard deviation of the numbers 2, 3, $$a$$ and 11 is 3.5, then which of the following is true?
A.
$$3{a^2} - 34a + 91 = 0$$
B.
$$3{a^2} - 23a + 44 = 0$$
C.
$$3{a^2} - 26a + 55 = 0$$
D.
$$3{a^2} - 32a + 84 = 0$$
Answer :
$$3{a^2} - 32a + 84 = 0$$
Solution :
$$\eqalign{
& \overline x = \frac{{2 + 3 + a + 11}}{4} \cr
& = \frac{a}{4} + 4 \cr
& \sigma = \sqrt {\sum {\frac{{x_i^2}}{n} - {{\left( {\overline x } \right)}^2}} } \cr
& \Rightarrow \,\,3.5 = \sqrt {\frac{{4 + 9 + {a^2} + 121}}{4} - {{\left( {\frac{a}{4} + 4} \right)}^2}} \cr
& \Rightarrow \,\,\frac{{49}}{4} = \frac{{4\left( {134 + {a^2}} \right) - \left( {{a^2} + 256 + 32a} \right)}}{{16}} \cr
& \Rightarrow 3{a^2} - 32a + 84 = 0 \cr} $$