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4 April, 09:29

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?

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Answers (1)
  1. 4 April, 09:37
    0
    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
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