Question

CASE STUDY CH.6 A spice manufacturer has a machine that fills bottles. The bottles are labeled 16 grams net weight so the company wants to have that much spice in each bottle. The company knows that just like any packaging process this packaging process is not perfect and that there will some variation in the amount filled. If the machine is set at exactly 16 grams and the normal distribution applies, then about half of the bottles will be underweight making the company vulnerable to bad publicity and potential lawsuits. To prevent underweight bottles, the manufacturer has set the mean a little higher than 16 grams. Based on their experience with the packaging machine, the company believes that the amount of spice in the bottle fits a normal distribution with a standard deviation of 0.2 grams. The company decides to set the machine to put an average 16.3 grams of spice in each bottle.

Answer 1

A spice manufacturer has a machine that fills bottles. The bottles are labeled 16 grams net weight so the company wants to have that much spice in each bottle. The company knows that just like any packaging process this packaging process is not perfect and that there will some variation in the amount filled. If the machine is set at exactly 16 grams and the normal distribution applies, then about half of the bottles will be underweight making the company vulnerable to bad publicity and potential lawsuits. To prevent underweight bottles, the manufacturer has set the mean a little higher than 16 grams. Based on their experience with the packaging machine, the company believes that the amount of spice in the bottle fits a normal distribution with a standard deviation of 0.2 grams. The company decides to set the machine to put an average 16.3 grams of spice in each bottle. Based on the above information answer the following questions:

1) What percentage of the bottles will be underweight? (5 Points)

Z value for 16, z =(16-16.3)/0.2 =-1.5

P( x <16) = P( z < -1.5) =0.0668

percentage of the bottles underweight = 6.68%

2) The company's lawyers says that the answer obtained in question 1 is too high. They insist that no more then 4% of the bottles can be underweight and the company needs to put a little more spice in each bottle. What mean setting do they need? (5 Points)

Z value for 4% level = 1.751

X = 16+1.751*0.2 = 16.3502

mean setting need = 16.35 grams.

3) The company CEO says that they do not want to give away too much free spice. She insists that the machine be set no higher than 16.2 grams (for the average) and still have only 4% underweight bottles as specified by the lawyers. This can be only accomplished by reducing the standard deviation. What standard deviation must the company achieve to meet the mandate from the CEO? (4 Points)

Z=(16.2-16)/sd = 1.751

Sd = 0.2/1.751 =0.1142

The required standard deviation = 0.1142

4) A disgruntled employee decides to set the machine to put an average 17.4 grams of spice in each bottle. What % of the bottles will be over weight (use standard deviation of 0.2 grams for this question)? (5 Points Hint: this question is similar to Question 1 but make sure you draw a diagram so as to answer this question correctly)

Z = (16-17.4)/0.2 = -7

P( z >-7) =1.0

All the bottles (100%) are over weight.