Question

Math

Posted 5 months ago

```
A vending machine that serves coffee pours a varying amount of liquid into a cup with varying mass. Assume that the masses of the liquid and the cup are independent.
Here are summary statistics for the masses of the liquid and the cups:
\begin{tabular}{lrr}
& Mean & Standard deviation \\
\hline Liquid & $\mu_{L}=250 \mathrm{~g}$ & $\sigma_{L}=20 \mathrm{~g}$ \\
Cup & $\mu_{C}=150 \mathrm{~g}$ & $\sigma_{C}=10 \mathrm{~g}$
\end{tabular}
Let $T=$ the total mass of a randomly selected cup filled with liquid from this machine.
Find the mean of $T$.
$\mu_{T}=$ $\square$ grams
```

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Answer from Sia

Posted 5 months ago

Solution by Steps

step 1

To find the mean of $T$, which is the total mass of a randomly selected cup filled with liquid, we need to add the mean mass of the liquid ($\mu_L$) and the mean mass of the cup ($\mu_C$)

step 2

Given that $\mu_L = 250$ grams and $\mu_C = 150$ grams, we calculate the mean of $T$ as follows: $\mu_T = \mu_L + \mu_C$

step 3

Substituting the given values into the equation from step 2, we get $\mu_T = 250 \text{ g} + 150 \text{ g}$

step 4

Calculating the sum, we find that $\mu_T = 400 \text{ g}$

Answer

$\mu_T = 400$ grams

Key Concept

Mean of Independent Random Variables

Explanation

The mean of the total mass $T$ is the sum of the means of the independent masses of the liquid and the cup.

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