Question

Company F sells fabrics known as fat quarters, which are rectangles of fabric created by cutting a yard of fabric into four pieces. Occasionally the manufacturing process results in a fabric defect. Let the random variable X represent the number of defects on a fat quarter created by Company F. The following table shows the probability distribution of X.

X 0 1 2 3 4 or more

Probability 0. 58 0. 23 0. 11 0. 05 0. 03

If a fat quarter has more than 2 defects, it cannot be sold and is discarded. Let the random variable Y represent the number of defects on a fat quarter that can be sold by Company F.

Determine the mean and standard deviation of Y. Show your work.

Company G also sells fat quarters. The mean and standard deviation of the number of defects on a fat quarter that can be sold by Company G are 0. 40 and 0. 66, respectively. The fat quarters sell for $5. 00 each, but are discounted by $1. 50 for each defect found.


(c) What are the mean and standard deviation of the selling price for the fat quarters sold by Company G?

Answer 1

(a) To determine the mean and standard deviation of Y, we need to find the expected value and standard deviation of the number of defects that can be sold by Company F. Since the number of defects that can be sold is equal to X if X is less than or equal to 2, and equal to 2 if X is greater than 2, we can use the following formula to find the expected value of Y:

E(Y) = P(X = 0) × 0 + P(X = 1) × 1 + P(X = 2) × 2 + P(X > 2) × 2

E(Y) = 0.58 × 0 + 0.23 × 1 + 0.11 × 2 + 0.03 × 2

E(Y) = 0.23 + 0.22 + 0.03

E(Y) = 0.48

To find the standard deviation of Y, we can use the following formula:

Var(Y) = P(X = 0) × (0 - E(Y))^2 + P(X = 1) × (1 - E(Y))^2 + P(X = 2) × (2 - E(Y))^2 + P(X > 2) × (2 - E(Y))^2

Var(Y) = 0.58 × (0 - 0.48)^2 + 0.23 × (1 - 0.48)^2 + 0.11 × (2 - 0.48)^2 + 0.03 × (2 - 0.48)^2

Var(Y) = 0.58 × 0.2304 + 0.23 × 0.1024 + 0.11 × 0.0304 + 0.03 × 0.0304

Var(Y) = 0.1333

The standard deviation of Y is the square root of the variance:

StdDev(Y) = √Var(Y)

StdDev(Y) = √0.1333

StdDev(Y) = 0.3663

So, the mean and standard deviation of Y are 0.48 and 0.3663, respectively.

(c) To find the mean and standard deviation of the selling price for the fat quarters sold by Company G, we need to find the expected value and standard deviation of the price, taking into account the discount for each defect. We can use the following formula to find the expected value of the price:

E(Price) = $5.00 - $1.50 × E(Defects)

E(Price) = $5.00 - $1.50 × 0.40

E(Price) = $5.00 - $0.60

E(Price) = $4.40

To find the standard deviation of the price, we can use the following formula:

StdDev(Price) = $1.50 × StdDev(Defects)

StdDev(Price) = $1.50 × 0.66

StdDev(Price) = $0.99

So, the mean and standard deviation of the selling price for the fat quarters sold by Company G are $4.40 and $0.99, respectively.