Answer:
The mean and standard devaition of W are 6 and 31 respectively.
Step-by-step explanation:
Given that:
W = 7 + 2X
Variance (W) = Variance(7 + 2X)
For a term constant term, the variance is zero.
So;
[tex]W = Var (2X)[/tex]
W = 4 × Var(X)
W = 4 × (SD (X))²
W = 4 × (3)²
W = 4 × 9
W = 36
Thus, Variance (W) = 36
The standard deviation of W = [tex]\sqrt{variance}[/tex]
W = [tex]\sqrt{36}[/tex]
W = 6
Also;
E(W) = E(7 +2 X)
E(W) = 7 + 2 × E(X)
where;
mean X = 12
E(W) = 7 + 2(12)
E(W) = 31
Thus, the mean and standard devaition of W are 6 and 31 respectively.
The mean and the standard deviation of W are 31 and 6 respectively.
The given parameters are:
[tex]\mathbf{\bar x_x = 12}[/tex]
[tex]\mathbf{\sigma_x = 3}[/tex]
The definition of W is given as:
[tex]\mathbf{W = 7 + 2X}[/tex]
This means that:
[tex]\mathbf{Var(W) = Var(7) + Var(2X)}[/tex]
So, we have:
[tex]\mathbf{Var(W) = 0 + Var(2X)}[/tex] --- the variance of a constant term is 0
[tex]\mathbf{Var(W) = Var(2X)}[/tex]
This gives
[tex]\mathbf{Var(W) = 2^2 \times Var(X)}[/tex]
We have:
[tex]\mathbf{Var(X) = \sigma^2_x}[/tex]
So, the equation becomes
[tex]\mathbf{Var(W) = 2^2 \times \sigma^2_x}[/tex]
[tex]\mathbf{Var(W) = 2^2 \times 3^2}[/tex]
[tex]\mathbf{Var(W) = 36}[/tex]
The standard deviation is then calculated as:
[tex]\mathbf{\sigma_W = \sqrt{Var(W)}}[/tex]
[tex]\mathbf{\sigma_W = \sqrt{36}}[/tex]
[tex]\mathbf{\sigma_W = 6}[/tex]
The mean is then calculated as:
[tex]\mathbf{E(W) = E(7 + 2X)}[/tex]
This gives
[tex]\mathbf{E(W) = 7 + 2E(X)}[/tex]
So, we have:
[tex]\mathbf{E(W) = 7 + 2 \times 12}[/tex]
[tex]\mathbf{E(W) = 31}[/tex]
Hence, the mean and the standard deviation of W are 31 and 6 respectively.
Read more about mean and standard deviation at:
https://brainly.com/question/10729938
