Ana is a dedicated Skee Ballplayer (see photo) who always rolls for the $50$-point slot. The probability distribution of Ana's score $X$on a single roll of the ball is shown below. You can check that ${\mu }_{\mathrm{X}}=23.8$and ${\sigma }_{\mathrm{X}}=12.63$.

(a) A player receives one ticket from the game for every $10$points scored. Make a graph of the probability distribution for the random variable $T=$number of tickets Ana gets on a randomly selected throw. Describe its shape.

(b) Find and interpret ${\mu }_{\mathrm{T}}$.

(c) Compute and interpret ${\sigma }_{\mathrm{T}}$.

The number of tickets is the number of points scored divided by$10$

$T=\frac{1}{10}X$

The probabilities for $T$are the same as the probabilities for $X$.

Create a probability histogram

The width of the bars has to be equal and the height has to be equal to probability.

Given

${\mu }_X=23.8$

${\sigma }_X=12.63$

The number of tickets is the number of points scored divided by$10$

$T=\frac{1}{10}X$

The probabilities for $T$are the same as the probabilities for $X$.

Properties mean and standard deviation:

${\mu }_{aX+b}=a{\mu }_X+b$

${\sigma }_{aX+b}=a{\sigma }_X$

Then we obtain for $T=\frac{1}{10}X$:

localid="1649909964609" $$\begin{gathered} {\mu }_T={\mu }_{1/10*X} \\ =\frac{1}{10}{\mu }_X \\ =\frac{1}{10}(23.8) \\ =2.38 \end{gathered}$$.

Given

${\mu }_X=23.8$

${\sigma }_X=12.63$

The number of tickets is the number of points scored divided by$10$

$T=\frac{1}{10}X$

The probabilities for $T$are the same as the probabilities for $X$.

Properties mean and standard deviation:

${\mu }_{aX+b}=a{\mu }_X+b$

${\sigma }_{aX+b}=a{\sigma }_X$

Then we obtain for $T=\frac{1}{10}X$:

localid="1649910001357" $$\begin{gathered} {\mu }_T={\mu }_{1/10*X} \\ =\frac{1}{10}{\mu }_X \\ =\frac{1}{10}(23.8) \\ =2.38 \end{gathered}$$

localid="1649910014351" $$\begin{gathered} {\sigma }_T={\sigma }_{1/10*X} \\ =\frac{1}{10}{\sigma }_X \\ =\frac{1}{10}(12.63) \\ =1.263 \end{gathered}$$

The number of tickets varies, on average, about $1.263$ticket about the mean of $2.38$tickets.