Question

How to solve this question

Answer 1

The variance method is as follows.

-Sum the squares of the values in data set, and then divide by the number of values in data set
- From that, subtract the square of the mean (add all values and divide by number of values in the data set)

Our variance is


`\displaystyle\sigma^2 = (2^2 + 5^2 + m^2)/(3) - \left((2 + 5 + m)/(3)\right)^2`

Since variance has to be 14, we set
`\sigma^2 = 14` and solve for m


`14= (4 + 25 + m^2)/(3) - \left((7 + m)/(3)\right)^2\ \Rightarrow \\ \\ 14 = (29)/(3) + (1)/(3)m^2 - (1)/(9)(7+m)^2 \\ \\ 14 = (29)/(3)+ (1)/(3)m^2 - (1)/(9)(49 + 14m + m^2) \\ \\ 14 = (29)/(3)+ (1)/(3)m^2 - (49)/(9)- (14)/(9)m- (1)/(9)m^2 \\ \\ 0 = (-88)/(9) -(14)/(9)m + (2)/(9)m^2`

quadratic formula



`m = \displaystyle(-b \pm √(b^2 -4ac))/(2a) \\ m = \frac{-(-(14)/(9)) \pm \sqrt{\left(-(14)/(9)\right)^2 - 4(2/9)(-88/9)} }{2(2/9)} \\ m = \frac{(14)/(9) \pm \sqrt{ (196)/(81) + (704)/(81) } }{(4)/(9) } \\ m = \frac{(14)/(9) \pm \sqrt{ (900)/(81) } }{(4)/(9) } \\ m = \frac{(14)/(9) \pm \sqrt{ (100)/(9) } }{(4)/(9) } \\ m = ((14)/(9) \pm (10)/(3) )/((4)/(9) ) \\ m = 11, -4`

-4 doesnt' work as it is not a positive integer

m = 11