Question

54. The Tri -State Pick $3$Refer to Exercise $42$. Suppose $365$
(a) Find the mean and standard deviation of your total winnings. Show your work.
(b) Interpret each of the values from (a) in context .

Short answer

(a) The mean and standard deviation of your total winnings are $-\$182.50$and $\$301.86$.

(b) The standard deviation shows that there will be loss of $\$182.50$ in an year which may average on an average by $\$5767$ in a year.

Step-by-step solution

Part (a) Step 1: Given information

Given in the question that the mean and standard deviation of the total winnings.

Part (a) Step 2: Calculate the mean and standard deviation of total winnings

According to the information, the population mean:

$\left(\mu \right)=\$0.50$
Population standard deviation:

$\left(\sigma \right)=\$15.80$.

Let's determine the mean and standard deviation of the total payoff as:

localid="1649873876116" $$\begin{gathered} {\mu }_P=365{\mu }_X-365 \\ =365\left(0.50\right)+0.50 \\ =-\mathrm{\$}182.50 \end{gathered}$$

Then,
localid="1649873879625" $$\begin{gathered} {\sigma }_p=\sqrt{{\sigma }_{X1}^2+{\sigma }_{X_2}^2+\dots +{\sigma }_{X365}^2} \\ =\sqrt{365\left(15.80\right)} \\ =\mathrm{\$}301.86 \end{gathered}$$

Part (b) Step 1: Given information

Interpret the values that determined from (a) in context.

Part (b) Step 2: Interpret the values

Let's consider the result from part (a):
$\mu =-\mathrm{\$}182.50$

$\sigma =\mathrm{\$}5767.00$

Hence, will lose about $\$182.50$in a year on average, which will vary on average by about $\$5767.00$in a year.