CASE STUDY CH.6 DROPBOX ASSIGNMENT A spi manufacturer has a machine that fills b

Question

CASE STUDY CH.6 DROPBOX ASSIGNMENT A spi 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) . 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) 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? (2 Points) Can you think of a practical way as to how the company can reduce the standard deviation for this bottle filling process? (1 Bonus Point) 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) 5) If the company asked you, as a consultant, to determine if its assumption about the amount of spice filled in the bottle is normally distributed is correct, what would you do List and explain 3 different steps you would take. to determine the normality of the data? (3 Points) Guidelines: As with past case studies submission only accepted via D2L Dropbox by due date to receive credit. No email submission. You can submit your work via MS Word or Excel. If you use the Excel templates to complete your work, please show ALL work and mark which answer is for which questions; not doing so will result in points deduction. Please write clearly and in complete sentences

Explanation / Answer

1. mean, mu is set as 16.3

std devn, sigma is 0.2

that means 50% is below 16.3 and 50% above 16.3

To find how much of the bottles are below 16

normalise 16 to z by (x-mu)/sigma

Probability of Z less than z

P( Z < (16-16.3)/0.2)

P(Z < -1.5).

Find the probability of z-value as-1.5 in the normal tables or using excel norm.s.dist(-1.5)

P (Z <-1.5) = 6.68%

2)

Let z be the new mean such than P( Z < z) is

To have P(Z < z) = 4%

P( Z < (x-mu)/sigma) = 4%

(x-mu)/sigma = z value of 4%

(16-mu)/0.2 =-1.75

mu = 16+1.75*0.2 = 16.35

If the mean is set as 16.35, then the bottles that will be less than 16 gms will be 4% or less.

3) Mean is set as 16.2, but the probability of bottles less than 16 should be 4%

P( Z < (16-16.2)/sigma) = 0.04

(16-16.2)/sigma = z value of 0.04

-0.2/sigma = -1.75

sigma = 0.1143

4)

mean is set to 17.4

the bottles that will be higher than 16 gms will be

P( Z > (16-17.4)/0.2)

= 1 - P(Z <(16-17.4)/0.2)

= 1 - P (Z < -7)

=1-1.27E-12

approx 1 or 100%

almost all of them will be overweight (close to 100%)

5)

Amount of spice over a long period of time is normally distributed.

1. Take a batch of 50 bottles (with spice) and calculate mean of the batch. This is 1 sample

2. Take 100 samples like the step 1.

3. Plot these 100 samples on a graph

It will be bell shaped with a mean and standard deviation.

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