Question

Questions T6.5 and T6.6 refer to the following setting. The weight of tomatoes chosen at random from a bin at the farmer’s market is a random variable with mean $\mu =10$ounces and standard deviation $\sigma =1$ounce. Suppose we pick four tomatoes at random from the bin and find their
total weight T.

T6.6. The random variable$T$ has a standard deviation (in ounces) of
(a) $0.25.$(b) $0.50.$(c) $0.71$. (d) $2$(e) $4$

Short answer

The random variable$T$ has a standard deviation of $2$ounces. So, the option(d) is correct.

Step-by-step solution

Given information

The farmer’s market is a random variable with mean $\mu =10$ounces and standard deviation $\sigma =1$ounce.

Explanation

Let, $X$(one tomato) is ${\mu }_X=10$and, ${\sigma }_X=1$
Then,$T$ is the random variable for the weight of four tomatoes:
$T=X_1+X_2+X_3+X_4$

And the properties mean and variance:

Note that $X$and $Y$are independent:

$$\begin{gathered} {\mu }_{X+Y}={\mu }_X+{\mu }_Y \\ {{\sigma }^2}_{X+Y}={{\sigma }^2}_X+{{\sigma }^2}_Y \end{gathered}$$

The mean and variance of $T$is:

$$\begin{gathered} {\mu }_T={\mu }_{X_1+X_2+X_3+X_4}=4{\mu }_X \\ =4\left(10\right)=40 \end{gathered}$$

ounces.

$$\begin{gathered} {\sigma }_T^2={\sigma }_{X_1+X_2+X_3+X_4}^2 \\ =4{\sigma }_X^2 \\ =4\left(1^2\right) \\ =4{\text{ounces}}^2 \end{gathered}$$

The standard deviation is the square root of the variance:

${\sigma }_T=\sqrt{{\sigma }_T^2}=\sqrt{4}=2\text{ounces}$