A company sells apples in packages of $4$. The weights of the individual
apples are normally distributed with a mean of $175g$, and a standard
deviation of $15g$. Assume that the weights of apples are independent
from each other. Let:
$$T = \text{the total weight of 4 randomly selected apples}$$
Find the probability that the total weight is more than $736g$.
Let $\text{X = the weight of 1 apple}$.
So here is my approach to find the mean and standard deviation of T (based on Impact of transforming (scaling and shifting) random variables):
\begin{aligned} \mu_T&=4\mu_X&=700 \end{aligned}
\begin{aligned} \sigma_T&=4\sigma_X&=60 \end{aligned}
But my standard deviation of T is wrong and I don't understand why. Here is what correct:
We can find the standard deviation of T by adding the variances of
the weights and taking the square root:
$$\sigma_T^2=15^2+15^2+15^2+15^2 =900$$
$$\Leftrightarrow \sigma_T =\sqrt{900} = 30$$
Please help me understand it, thank you.
Best Answer
If you have independent events X, Y, the variance of these events is simply the sum of variances. You can calculate the variance first and after that calculate the standard deviation. Eg.
$$Var(T) = Var(X_1) + Var(X_2) + Var(X_3) + Var(X_4)$$ $$Var(T) = 15^2 + 15^2 + 15^2 + 15 ^ 2$$ And standard deviation is the square root of this. $$\sigma(T) = {\sqrt{Var(T)}} = \sqrt{900} = 30$$
The reason is that you cannot add standard deviations but you can add variances.