Question

a toy store employee counted the number of toys in each bin in the sale section. x is the number of toys that a randomly chosen bin had. what is the variance of x? write your answer as a decimal.

Explanation:

Step1: Calculate the total number of bins

$$ 7 + 2+4 + 4+3=20 $$

Step2: Calculate the probability distribution

For \(x = 23\), \(P(X = 23)=\frac{7}{20}=0.35\)
For \(x = 28\), \(P(X = 28)=\frac{2}{20}=0.1\)
For \(x = 34\), \(P(X = 34)=\frac{4}{20}=0.2\)
For \(x = 55\), \(P(X = 55)=\frac{4}{20}=0.2\)
For \(x = 61\), \(P(X = 61)=\frac{3}{20}=0.15\)

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

$$\begin{align*} E(X)&=(23\times0.35)+(28\times0.1)+(34\times0.2)+(55\times0.2)+(61\times0.15)\\ &=8.05 + 2.8+6.8+11+9.15\\ &=37.8 \end{align*}$$

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

$$\begin{align*} E(X^{2})&=(23^{2}\times0.35)+(28^{2}\times0.1)+(34^{2}\times0.2)+(55^{2}\times0.2)+(61^{2}\times0.15)\\ &=(529\times0.35)+(784\times0.1)+(1156\times0.2)+(3025\times0.2)+(3721\times0.15)\\ &=185.15+78.4 + 231.2+605+558.15\\ &=1658 \end{align*}$$

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

Using the formula \(Var(X)=E(X^{2})-[E(X)]^{2}\)

$$ Var(X)=1658-(37.8)^{2}=1658 - 1428.84=229.16 $$

Answer:

\(229.16\)