Question

Get on the boat! A small ferry runs every half hour from one side of a large river to the other. The probability distribution for the random variable Y = money collected on a randomly selected ferry trip is shown here. From Exercise 7, $\mu Y=\$19.35$. Calculate and interpret the standard deviation of Y.

Short answer

The standard deviation is$\sigma =6.4286$.

Step-by-step solution

Step 1. Given information.

The given information is:

Step 2. Find and interpret the standard deviation of Y. 

The given mean is 19.35.

The predicted value of the squared departure from the mean is known as the variance:

$$\begin{gathered} {\sigma }^2=\sum {\left(x-\mu \right)}^{2}P\left(x\right) \\ ={\left(0-19.35\right)}^2\times \left(0.02\right)+{\left(5-19.35\right)}^2\times \left(0.05\right)+{\left(10-19.35\right)}^2\times \left(0.08\right)+ \\ {\left(15-19.35\right)}^2\times \left(0.16\right)+{\left(20-19.35\right)}^2\times \left(0.27\right)+{\left(25-19.35\right)}^2\times \left(0.42\right) \\ =41.3275 \end{gathered}$$

The standard deviation is:

$$\begin{gathered} \sigma =\sqrt{41.3275} \\ \approx 6.4286 \end{gathered}$$

The amount of money collected on a randomly selected boat journey differs by $6.4286 from the mean amount collected ($19.35) on average.