a pottery factory kept track of the number of broken plates per shipment last week. x is the number of broken plates that a randomly chosen shipment had. what is the variance of x? write your answer as a decimal.

Question

Accepted Answer

# Explanation:
## Step1: Calculate the total number of shipments
Sum up the number of shipments: $4 + 4+2 + 4+3 + 3=20$
## Step2: Calculate the probability distribution
For $x = 147$, $P(X = 147)=\frac{4}{20}=0.2$
For $x = 171$, $P(X = 171)=\frac{4}{20}=0.2$
For $x = 348$, $P(X = 348)=\frac{2}{20}=0.1$
For $x = 521$, $P(X = 521)=\frac{4}{20}=0.2$
For $x = 767$, $P(X = 767)=\frac{3}{20}=0.15$
For $x = 922$, $P(X = 922)=\frac{3}{20}=0.15$
## Step3: Calculate the expected value $E(X)$
$$
\begin{align*}
E(X)&=(147\times0.2)+(171\times0.2)+(348\times0.1)+(521\times0.2)+(767\times0.15)+(922\times0.15)\
&=29.4 + 34.2+34.8+104.2+115.05+138.3\
&=455.95
\end{align*}
$$
## Step4: Calculate $E(X^{2})$
$$
\begin{align*}
E(X^{2})&=(147^{2}\times0.2)+(171^{2}\times0.2)+(348^{2}\times0.1)+(521^{2}\times0.2)+(767^{2}\times0.15)+(922^{2}\times0.15)\
&=(21609\times0.2)+(29241\times0.2)+(121104\times0.1)+(271441\times0.2)+(588289\times0.15)+(850084\times0.15)\
&=4321.8+5848.2 + 12110.4+54288.2+88243.35+127512.6\
&=292324.55
\end{align*}
$$
## Step5: Calculate the variance $Var(X)$
Using the formula $Var(X)=E(X^{2})-(E(X))^{2}$
$$
\begin{align*}
Var(X)&=292324.55-(455.95)^{2}\
&=292324.55 - 207891.4025\
&=84433.1475
\end{align*}
$$

# Answer:
$84433.15$

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Question

Explanation:

Step1: Calculate the total number of shipments

Step2: Calculate the probability distribution

Step3: Calculate the expected value \(E(X)\)

Step4: Calculate \(E(X^{2})\)

Step5: Calculate the variance \(Var(X)\)

Answer: