Question

AP2.20 A grocery chain runs a prize game by giving each customer a ticket that may win a prize when the box is scratched off. Printed on the ticket is a dollar value ( \( 500, \) 100, \(25) or the statement "This ticket is not a winner." Monetary prizes can be redeemed for groceries at the store. Here is the probability distribution of the amount won on a randomly selected ticket:

Which of the following are the mean and standard deviation, respectively, of the winnings?

a. \) 15.00, \( 2900.00

b.\) 15.00, \( 53.85

c. \) 15.00, \( 26.93

d. \) 156.25,\( 53.85

e. \) 156.25, $ 26.93

Short answer

b.$ 15.00, $ 53.85

Step-by-step solution

Given Information

Formula used:

$$\begin{gathered} \mu =\sum x\times p\left(x\right) \\ {\sigma }^2=\sum (x-\mu {)}^2\times p(x) \\ \sigma =\sqrt{{\sigma }^2} \end{gathered}$$

Explanation for correct option

$\mu =500\times 0.01+100\times 0.05+25\times 0.20+0\times 0.74=15$

Variance

$$\begin{gathered} {\sigma }^2=(500-15{)}^2\times 0.01+(100-15{)}^2\times 0.05 \\ +(25-15{)}^2\times 0.20+(0-15{)}^2\times 0.74 \\ =2900 \end{gathered}$$

Standard deviation

$\sigma =\sqrt{2900}=53.85$

Hence, the correct option is (b)

Explanation for incorrect option

a. $ 15.00, $ 2900.00 is not the answer.

c. $ 15.00, $ 26.93 is not the answer.

d. $ 156.25,$ 53.85 is not the answer.

e. $ 156.25, $ 26.93 is not the answer.