A random sample of size n1 = 16 is selected from a normal population with a mean of 75 and a standard deviation of 8. A second random sample of size n2 = 9 is taken from another normal population with mean 70 and standard deviation 12.Observe that the populations being sampled from have different variances. In view of that, are equal samples sizes best? If not, then what would be? Hint: let m be the total sample size with n1 = cm and n2 = (1−c)m. Determine Var (X̅1- X̅2 )as a function of c and show that it is minimized with c= σ1/(σ 1+σ2).

I starting with Var(X̅1- X̅2) = (σ²1∕cm) +(σ²2∕(1-c)m)

= ((1-c)σ²1+σ²2 c)/mc(1-c)

Then for the variance to be minimum,

Var (X̅1- X̅2) = 0 ,

Or, ((1-c)σ²1+σ²2 c)/mc(1-c)= 0

Or, ((1-c)σ²1+σ²2 c) =0

Or, (σ²1 -c σ²1 +σ²2 c) =0

Or, c= σ²1/( σ²1 -σ²2)

I could not figure out how to solve this question . please guide me on it.

A random sample of size n1 = 16 is selected from a normal population with a mean of 75 and a standard deviation of 8. A second random sample of size n2 = 9 is taken from another normal population with mean 70 and standard deviation 12.Observe that the populations being sampled from have different variances. In view of that, are equal samples sizes best? If not, then what would be? Hint: let m be the total sample size with n1 = cm and n2 = (1−c)m. Determine Var (X̅1- X̅2 )as a function of c and show that it is minimized with c= σ1/(σ 1+σ2).

I starting with Var(X̅1- X̅2) = (σ²1∕cm) +(σ²2∕(1-c)m)

= ((1-c)σ²1+σ²2 c)/mc(1-c)

Then for the variance to be minimum,

Var (X̅1- X̅2) = var X1 +var X2

= ((1-c)σ²1+σ²2 c)/m(c+(1-c))

= ((1-c)σ²1+σ²2 c)/m

=(σ²1 -σ²1c +σ²2 c)/m

I could not figure out how to solve this question . please guide me on it.