The mean and standard deviation of \(\mathrm{x}\) is 40 and 4 respectively the mean and standard deviation of \([(x-40) / 4]\) is (a) 1,0 (b) 1,1 (c) 0,1 (d) \(0,-1\)

To find the mean of the transformed variable \([(x-40) / 4]\), we will calculate the expected value of this expression. Since the mean of \(x\) is given as 40, we have: \[E[(x-40)/4] = \frac{1}{4}E[(x-40)] = \frac{1}{4}(E[x]-40)\] Now, substitute the given mean of \(x\), 40, into the equation: \[\frac{1}{4}(40-40) = \frac{1}{4}(0) = 0\] So, the mean of the transformed variable is 0.

To find the standard deviation of the transformed variable \([(x-40) / 4]\), we will first find the variance of this expression. The variance of a constant times a variable is equal to the square of the constant times the variance of the variable: \[\text{Var}[(x-40)/4] = \left(\frac{1}{4}\right)^2\text{Var}[(x-40)]\] Since adding a constant doesn't change the variance, \[\text{Var}[(x-40)/4] = \left(\frac{1}{4}\right)^2\text{Var}[x]\] Now, substitute the given standard deviation of \(x\), 4, into the equation (recall that the variance is the square of the standard deviation): \[\left(\frac{1}{4}\right)^2 (4^2) = (1/16)(16) = 1\] Now, we find the standard deviation by taking the square root of the variance: \[\sqrt{1} = 1\] So, the standard deviation of the transformed variable is 1.

We found that the mean of the transformed variable is 0 and its standard deviation is 1. By comparing our results with the given options, we see that the answer corresponds to option (c) 0, 1.