Question

multiple choice:

1. a company that manufactures and sells men’s suits makes the jackets at the east coast site and the pants at the west coast site. if the two sites produce suits with a number of defects reflected by $\mu_j = 0.002$, $\sigma_j = 0.15$ and $\mu_p = 0.005$, $\sigma_p = 0.18$, what is the mean and standard deviation for the number of total defects for suits produced by this company?

a.) $\mu = 0.00035$ and $\sigma = 0.165$
b.) $\mu = 0.00035$ and $\sigma = 0.0549$
c.) $\mu = 0.007$ and $\sigma = 0.0549$
d.) $\mu = 0.007$ and $\sigma = 0.2343$
e.) $\mu = 0.007$ and $\sigma = 0.33$

1. a game of chance involves spinning a fair 6 sided spinner seen to the right. if the spinner lands on blue, you win $4. if it lands on red, you win $0.50. if it lands on any other color, you win nothing. if the cost to play the game is $1, what are a player’s expected profit?

a.) lose $0.08
b.) lose $0.25
c.) win $0.08
d.) win $0.25
e.) win $0.75

1. the table below represents the winnings from a card game. let the random variable x represent the quantity you win from playing the game. what is the expected value of x?

x = winnings	$1	$5	$10	$20	$50

a.) $1.00
b.) $3.90
c.) $4.00
d.) $10.00
e.) $17.20

1. your school is holding a raffle. the school claims you can win $2, $5, $15 or a mystery amount. the likelihood of winning each of the prizes from playing the raffle are displayed in the table below.

x = winnings from the raffle

x = prize amount	$2	$5	$15	mystery amount

the expected value of x is $9.10. what is the mystery amount?
a.) $8
b.) $8.52
c.) $10
d.) $12
e.) $14.40

Explanation:

Question 1

Step1: Find the mean of total defects

The mean of the sum of two independent random variables is the sum of their means. So, \(\mu=\mu_J + \mu_P\). Given \(\mu_J = 0.002\) and \(\mu_P=0.005\), we have \(\mu=0.002 + 0.005=0.007\).

Step2: Find the standard deviation of total defects

For independent random variables, the standard deviation of the sum is \(\sigma=\sqrt{\sigma_J^2+\sigma_P^2}\). Given \(\sigma_J = 0.15\) and \(\sigma_P = 0.18\), we calculate \(\sigma_J^2=0.15^2 = 0.0225\) and \(\sigma_P^2=0.18^2=0.0324\). Then \(\sigma=\sqrt{0.0225 + 0.0324}=\sqrt{0.0549}\approx0.2343\).

Step1: Determine probabilities

The spinner is 6 - sided, so the probability of landing on any color is \(\frac{1}{6}\). Probability of blue (\(P(B)\)) is \(\frac{1}{6}\), red (\(P(R)\)) is \(\frac{1}{6}\), and other colors (\(P(O)\)) is \(\frac{4}{6}=\frac{2}{3}\).

Step2: Calculate expected winning

Expected winning \(E(W)=4\times\frac{1}{6}+0.5\times\frac{1}{6}+0\times\frac{2}{3}\). Calculate: \(4\times\frac{1}{6}=\frac{4}{6}\approx0.6667\), \(0.5\times\frac{1}{6}=\frac{0.5}{6}\approx0.0833\), so \(E(W)=\frac{4 + 0.5}{6}=\frac{4.5}{6}=0.75\).

Step3: Calculate expected profit

Profit = Winning - Cost. Cost is \(\$1\), so expected profit \(E(P)=E(W)-1=0.75 - 1=- 0.25\) (lose \(\$0.25\)).

Step1: Recall expected value formula

The expected value \(E(X)=\sum x\cdot P(x)\).

Step2: Calculate each term

For \(x = 1\), \(x\cdot P(x)=1\times0.60 = 0.60\); for \(x = 5\), \(x\cdot P(x)=5\times0.30 = 1.50\); for \(x = 10\), \(x\cdot P(x)=10\times0.05 = 0.50\); for \(x = 20\), \(x\cdot P(x)=20\times0.04 = 0.80\); for \(x = 50\), \(x\cdot P(x)=50\times0.01 = 0.50\).

Step3: Sum the terms

\(E(X)=0.60+1.50 + 0.50+0.80 + 0.50=3.90\).

Answer:

d.) \(\mu = 0.007\) and \(\sigma=0.2343\)

Question 2