The amount of toothpaste in a tube is normally distributed with a mean of 6.5 ounces and a standard

deviation of 0.8 ounces. The cost of producing each tube is 50 cents. If in a quality control examination a

tube is found to weigh less than 6 ounces, it is to be refilled to the mean value at a cost of 20 cents per tube.

On the other hand, if the tube weighs more than 7 ounces, the company loses a profit of 5 cents per tube.

Assume 1,000 tubes are examined.

a). How many tubes will be found to contain less than 6 ounces? In that case, what will be the total cost of the

refill?

b) How many tubes will be found to contain more than 7 ounces? In that case, what will be the amount of

profit lost?

a) 266 tubes, TC_r = $53.2

b) 266 tubes, T. Loss = $13.30

Step-by-step explanation:

Given:

- The sample size of tubes n = 1,000 tubes

- The mean of the sample u = 6.5 oz

- The standard deviation of the sample s. d = 0.8 oz

- Cost of manufacturing a tube C_t = 50 cents

- Cost of refilling a tube C_r = 20 cents

- Profit loss per tube Loss = 5 cents

Find:

a). How many tubes will be found to contain less than 6 ounces? In that case, what will be the total cost of the refill?

b) How many tubes will be found to contain more than 7 ounces? In that case, what will be the amount of profit lost?

Solution:

- First we will compute the probability of tube containing less than 6 oz.

- Declaring X : The amount of toothpaste.

Where, X ~ N (6.5, 0.8)

- We need to compute P (X < 6 oz) ?

Compute the Z-score value:

P (X < 6 oz) = P (Z < (6 - 6.5) / 0.8) = P (Z < - 0.625)

Use the Z table to find the probability:

P (X < 6 oz) = P (Z < - 0.625) = 0.266

- The probability that it lies below 6 ounces. The total sample size is n = 1000.

The number of tubes with X < 6 ounces = 1000 * P (X < 6 oz)

= 1000*0.266 = 266 tubes.

- The total cost of refill:

TC_r = C_f * (number of tubes with X < 6)

TC_r = 20*266 = 5320 cents = $53.2

- We need to compute P (X > 7 oz) ?

Compute the Z-score value:

P (X > 7 oz) = P (Z > (7 - 6.5) / 0.8) = P (Z < 0.625)

Use the Z table to find the probability:

P (X > 7 oz) = P (Z > 0.625) = 0.266

- The probability that it lies above 7 ounces. The total sample size is n = 1000.

The number of tubes with X > 7 ounces = 1000 * P (X > 7 oz)

= 1000*0.266 = 266 tubes.

- The total cost of refill:

T. Loss = Loss * (number of tubes with X > 7)

T. Loss = 5*266 = 1330 cents = $13.30