 Hello and welcome to the session, the given question says the mean and variance of 7 observations are 8, 10, 16 respectively. If 5 observations are 2, 4, 10, 12 and 14, find the remaining 2 observations. So let's start with the solution and the given 5 observations are 2, 4, 10, 12 and 14 and let the remaining 2 observations be x and y. Now we are given that mean of these 7 observations is 8 and mean which is denoted by x bar is given by summation xi, i going from 1 to 7 upon 7, mean of 7 observation is given by this formula where xi 1 to 7 denotes these 7 observations where x 1 is 2, x 2 is 4, x 3 is 10, x 4 is 12, x 5 is 14, x 6 is x and x 7 is y. Now we are given that mean of these observations as 8. So 8 is equal to 2 plus 4 plus 10 plus 12 plus 14 plus x plus y 1 7 which implies 56 is equal to on adding we get 42 plus x plus y or x plus sorry this is plus is equal to 56 minus 42 which is equal to 14. So this implies x plus y is equal to 14. Let this be equation number 1. Also we are given that the variance of these 7 observations is equal to 16 and the formula to calculate the variance of 7 observation is given by summation i goes from 1 to 7 xi minus x bar whole square upon 7. Now let us make a table. So this is the required table in the first column we are writing the values of xi that is x 1, x 2, x 3, x 4, x 5, x 6 and x 7. In the second row we shall find the value of xi minus x bar and here x bar is equal to the mean of the 7 observations which is 8. Now let us find x 1 minus x bar that is 2 minus 8 gives minus 6 and 4 minus 8 gives minus 4, 10 minus 8 gives 2, 12 minus 8 gives 4, 14 minus 8 gives 6, then x minus 8 and y minus 8. Now let us find the squares minus 6 square is 36, minus 4 square is 16, 2 square is 4, 4 square is 16, 6 square is 36 and square of x minus 8 is x square minus 16 x plus 64 and square of y minus 8 is y square minus 16 y plus 64. Now let us find the sum this is equal to x square plus y square minus 16 x minus 16 y plus 236 on adding these 7 we get this value. Now let us substitute the value in the formula of this variance. Variance is given to us as 16. So 16 is equal to summation i going from 1 to 7 xi minus x bar is equal to x square plus y square minus 16 x minus 16 y plus 236 upon 7 or this is equal to 16 into 7 taking 236 on the left hand side. So equal to x square plus y square minus 16 x minus 16 y and on simplifying this 16 into 7 is 112 minus 236 is minus 124 this is equal to x square plus y square minus 16 into x plus y. Now some equation 1 we know that x plus y is equal to 14. So we have minus 124 and on multiplying 16 14 we get 224. So taking it on the right hand left hand side from the right hand side we have x square plus y square. So this implies x square plus y square is equal to 100. Now on adding 2 x y on both the sides we have 100 plus 2 x y on the right hand side. This is the formula of x plus y whole square and on the right hand side we have 100 plus 2 x y. Now x plus y is 14 and 14 square is 196 is equal to 100 plus 2 x y this implies 2 x y is equal to 96. Let us denote this by equation number 2. Now x minus y whole square is given by x square plus y square minus 2 x y and x square plus y square is 100 and minus value of 2 x y is 96 from equation number 2 we get 4. So this implies x minus y is equal to root over 4 and its value is plus minus 2. Let us denote this by equation number 3. And now from equation 1 which is x plus y is equal to 14 this implies y is equal to 14 minus x. So equation 3 can be written as x minus 14 minus x is equal to plus minus 2 or we have 2 x is equal to 14 plus minus 2 or 2 x is equal to 14 minus 2 is 12 and 2 x is equal to 14 plus 2 is 16. So this implies x is equal to 6 or x is equal to 8. Now for x is equal to 6 y is equal to 14 minus x that is 14 minus 6 which is equal to 8 and for x is equal to 8 y is equal to 14 minus 8 and this is equal to 6. So if x is 6 y is 8 and if x is 8 y is 6 so we can say that the remaining 2 observations are this completes the session hope you have understood it take care and have a good day.