Question

Victoria parks her car at the same garage every time she goes to work. Because she stays at work for different lengths of time each day, the fee the parking garage charges on a randomly selected day is a random variable, $G$. The table gives the probability distribution of $G.$You can check that ${\mu }_G=\backslash (14$and ${\sigma }_G=\backslash )2.74$.

In addition to the garage’s fee, the city charges a $\$3$use tax each time Victoria parks her car. Let $T=$the total amount of money she pays on a randomly selected day.

a. Make a graph of the probability distribution of $T$. Describe its shape.

b. Find and interpret $\mu T$.

c. Calculate and interpret $\sigma T$.

Short answer

a. The distribution is not symmetric, with the tallest bars in the middle at $X=18$, a single peak with the highest likelihood, and the lowest bars at $X=23$, with the lowest probability.

b. When the constant value is added to each data point, the center of the distribution is enlarged by $({\mu }_G+3)$ and the total amount is increased by $\$3$, resulting in $\$17$average money being randomly selected.

c. When a constant value is added to all data, it has no effect on the spread of the distribution, hence money for a random selection ranges on average $\$2.74$about the mean $=\$17.$

Step-by-step solution

Part(a) Step 1 : Given Information

Given table:

Part(a) Step 2 : Simplification

The city charges a $\$3$use tax each time Victoria parks her car.

So,$X=Y+3$

The required graph is :

Part(b) Step 1 : Given Information

Given table :

Part(b) Step 2 : Simplification

$Y$stands for garage cost. $\$3$must be paid in addition to the garage cost, bringing the total price to role="math" localid="1653997585789" $\$3$

$\Rightarrow X=Y+3.$

When the constant is added to each data value, the distribution's center is also enhanced by that constant value.

$$\begin{gathered} {\mu }_T=\mu G+3 \\ =14+3Mr \\ =\mathbf{17} \end{gathered}$$

Part(c) Step 1 : Given Information

Given table :

Part(c) Step 2 : Simplification

$Y$stands for garage cost. The garage fee must be paid in addition by $\$3$, bringing the total price to $\$3$.

$X=Y+3$

When you add the constant to every data value, the spread of the distribution does not change; it stays the same.

${\sigma }_x={\sigma }_y=2.74$