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 $T$.
T6.5. The random variable $T$ has a mean of
(a) $2.5$ ounces.
(b) $4$ounces.
(c) $10$ounces

(d) $40$ounces.

(e) $41$ounces

Short answer

The random variable $T$has a mean of $40$ounces. So, option (d) is correct.

Step-by-step solution

Given information

The weight of tomatoes is a random variable with mean $\mu =10$ounces and standard deviation $\sigma =1$ ounce.

Explanation

Let,$X$(one tomato) as:

${\mu }_X=10$, and ${\sigma }_X=1$

Then,$T$is the random variable for the weight of four tomatoes:

$T=X_1+X_2+X_3+X_4$

The mean and variance (where, $X$and$Y$are independent):

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

The mean and variance of $T$is then:

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

ounces

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