 Hello and welcome to the session the given question says, the mean and variance of 8 observations are 9 and 9.25 respectively. If 6 of the observations are 6, 7, 10, 12, 12 and 13 find the remaining two observations. So let's start with the solution and the given 6 observations are 6, 7, 10, 12, and 13 and let the remaining two observations which we have to find be x and y. Now we are given that mean of 8 observations is 9. So this implies mean which is denoted by x bar is equal to summation i going from 1 to 8 xi upon 8 and here the observations are 6, 7, 10, 12, 12, 13, x and y. So we have given the mean as 9 is equal to 6 plus 7 plus 10 plus 12 plus 12 plus 13 plus x plus y upon 8. Since here xi i going from 1 to 8 denotes the observations which are given to us and x and y. Now on simplifying we have 8 nines are 72 is equal to 60 plus x plus y or we have x plus y is equal to 12. Let us denote this equation by equation number one. Also we are given the variance which is equal to 9.25. So first we have to find the value of xi minus x bar where xi is x1, x2 up to x8, x1 is 6, x2 is 7, x3 is 10, x4 is 12, x5 is 12, x6 is 13, x7 is x and x8 is y. So first let us find the variance which is equal to summation as i goes from 1 to 8 xi minus x bar upon 8. Let us make a table. So this is the table in which first column denotes the xi that is the observations from 1 to 8 and the second column we shall find the value of xi minus x bar and the third column we shall find xi minus x bar whole square. Since to find the variance or formalize summation as i goes from 1 to 8 xi minus x bar sorry it is whole square upon 8. Now let us find the value of xi minus x bar. Now first element which is x1 is 6, so x1 minus x bar which is equal to 9 gives minus 3, 7 minus 9 is minus 2, 10 minus 9 is 1, 12 minus 9 is 3, again 12 minus 9 is 3, 13 minus 9 is 4, here we have x minus 9 and y minus 9. Now let us find the squares, square of minus 3 is 9, square of minus 2 is 4, square of 1 is 1, 3 is 9, square of this 3 is also 9, square of 4 is 16, square of x minus 9 is x square minus 18x plus 81 and square of 5 minus 9 is y square minus 18y plus 81. Now let us find summation of i goes from 1 to an xi minus x bar whole square. So that we will get on adding these 8 values and on adding we get x square plus y square minus 18x minus 18y plus 210. So now let us substitute the value of summation i goes from 1 to 8 xi minus x bar whole square in this formula. Now we are given that variance of these 8 observations is 9.25, so this is equal to x square plus y square minus 18x minus 18y plus 210 upon 8 and this is equal to 74 minus 210 is equal to x square plus y square minus 18x minus 18y which is further equal to minus 136 is equal to x square plus y square minus 18 times of x plus y and the value of x plus y from equation number 1 we know that it is equal to 12. So we further have this since x plus y is equal to 12 from equation number 1. Now simplifying it further we have minus 136 plus on multiplying 18 with 12 we get 216. So it is equal to x square plus y square or this is further equal to 80 is equal to x square plus y square. Now adding 2xy on both the sides this further implies that 80 plus 2xy is equal to x plus y whole square. And now again substituting the value of x plus y on the right hand side we have 144. So this implies 2xy is equal to 144 minus 80 which is equal to 64 or we have xy is equal to 32. Now x minus y whole square is given by x square plus y square minus 2xy and x square plus y square is equal to 80 minus 2 into xy that is 32 which further implies 80 minus 64 which is equal to 16. So this implies x minus y is equal to plus minus 4. Let this be equation number 2. Now from equation 1 which is x plus y is equal to 12 we get y is equal to 12 minus x. Now substituting the value of y which is equal to 12 minus x in equation number 2 we get x minus 12 minus x is equal to plus minus 4 or we have 2x minus 12 is equal to plus minus 4 or 2x is equal to 12 plus minus 4 which implies 2x is equal to 16 or 2x is equal to 8 which further implies that x is equal to 8 or x is equal to 4. Now from equation 1 we have y is equal to 12 minus x. Now for x is equal to 8 y is equal to 4 for x is equal to 4 y is equal to 12 minus 4 is equal to 8. So thus we have when x is equal to 8 then y is equal to 4 and when x is equal to 4 y is equal to 8 thus the remaining 2 observations are 4 and 8. So this completes the session. Hope you have understood it. Take care and have a good day.