Homework SourseThe amount of toothpaste in a tube is normally distributed w
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. If 100 tubes were examined, a. how many tubes will be found to contain less than 6 ounces?
b. in that case, what will be the total cost of the refill?
c. how many tubes will be found to contain more than 7 ounces and in that case what will be the amount of profits lost?
Solution
a)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 6
u = mean = 6.5
s = standard deviation = 0.8
Thus,
z = (x - u) / s = -0.625
Thus, using a table/technology, the left tailed area of this is
P(z < -0.625 ) = 0.265985529
Thus, there will be 0.265985529*100 = 26.5985529 tubes. [ANSWER, 26.5985529, please round off if that\'s what you do in class]
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b)
Thus, as the cost is 0.20 per tube,
Total cost of refil = 26.5985529 * 0.20 = $5.31971058 [ANSWER]
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c)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 7
u = mean = 6.5
s = standard deviation = 0.8
Thus,
z = (x - u) / s = 0.625
Thus, using a table/technology, the right tailed area of this is
P(z > 0.625 ) = 0.265985529 [ANSWER]
Thus, 100*0.265985529 = 26.5985529 will be more than 7 oz. [ANSWER]
A profit of 26.5985529*0.05= $1.329927645 will be lost. [ANSWER]
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Please round off the answers accordingly as to how you do it in your class.
