Trabajo_entregar_2012
Transcript of Trabajo_entregar_2012
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UNIVERSIDAD CARLOS III DE MADRID
Grados en Ingeniera Industrial(Mecnica, Elctrica, Electrnica Industrial y Automtica y Tecnologa Industrial)
Continuous Assessment Assignment (2012)
Apellidos:
Nombre:Grupo pequeo:
Only individual and original work will be marked. You mustsign the box to confirm that all of this assignment is your own
original work otherwise it will not be marked.Firma:
THIS ASSIGNMENT MUST BE PRINTED AND THEN ALL ANSWERS MUST BE
HANDWRITTEN IN THE BOXES PROVIDED
ONLY ANSWERS GIVEN COMPLETELY WITHIN THE BOXES BELOW EACHQUESTION PART WILL BE CONSIDERED
ALL SHEETS MUST BE HANDWRITTEN ORIGINALS - NO COPIES WILL BE
ACCEPTED
ALL SHEETS MUST BE STAPLED TOGETHER
ASSIGNMENTS THAT ARE NOT HANDED IN BY THE GIVEN DEADLINE WILL
BE PENALISED AND MAY NOT BE ACCEPTED
Question 1. (4 points)A quality control procedure was implemented in a factory that manufactures porcelain mugs.
Nsamples of mugs were taken, each with size n, whereN= 20 and n = 50. The number of
defects in each sample is recorded below.
Sample 1 2 3 4 5 6 7 8 9 10
No.
defects8 1 18 18 9 9 6 18 7 2
Sample 11 12 13 14 15 16 17 18 19 20
No.
defects10 7 4 8 1 3 18 19 11 1
Sum of No. defects = 178
(N.B. This entire question must be performed by hand, and not using StatGraphics. All
working and formulae used must be shown. NO marks will be awarded for simply stating the
answers.)
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a) Calculate the capacity of the process.
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b) Using the average value of probability of defects from the process (when under control),
calculate the probability that if ten mugs are taken at random, that three will be defective.
c) In the monitorization process, a sample size of 20 will be used. Calculate the required
values for the 3 sigma limits.
Question 2. (3 points)Two normal six-sided dice each have two faces painted red, two faces painted green and two
faces painted yellow. An experiment is defined as throwing both die.
a) Draw the sample space for the experiment.
b) Clearly denote the following events on the sample space above, and calculate the
probability of each event (A, B and C) below:
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i) A: exactly one yellow face
ii) B: at least one yellow face
iii) C: two yellow faces
c) Calculate the probability that two yellow faces are obtained if you know that one die has a
yellow face. Express this probability using formal notation in terms of the events defined, andderive its value.
d) You are offered the following game:
You pay 5 to play the game. If you get only one yellow face you will get paid 8,however if you get two yellow faces you will get paid 15. Otherwise, you are paid
nothing.
Should you accept to play the game? Justify your answer fully, and interpret the result of any
calculation performed.
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Question 3. (3 points)You are advising a friend who is an engineer and who works in a food company. As part of its
product range, the company manufactures tea bags. The weight of each finished bag is
measured before packing. Previous experience shows that the mean of the process is 1.6g with
a standard deviation of 0.2g, and the weight is normally distributed.
a) What are the limits (centred on the mean value of the process) that will contain the weight
of 95% of tea bags? (Tea bags whose weight is outside these limits are termed defective.)
b) You explain to your friend the meaning of the 95% weight limits. He takes a random
sample of 20 tea bags and finds two tea bags outside of the 95% weight limits (i.e., twodefective tea bags). One of his colleagues takes another random sample of 20 tea bags and
finds none out of tolerance. Neither of these results corresponds to the figure used in your
explanation. How do you explain this to him?
c) In order to regain his confidence in your knowledge of statistics, you explain that you can
model the process of taking tea bags from the production line. You tell him that statisticians
refer to this as an experiment of taking 20 tea bags at random, and the event of selecting a
defective tea bag as being a success.
State the probability model you would use to model the experiment of taking 20 tea bags from
the production line, which have a constant probability of being outside the 95% weight limits.
UseXto represent the number of successes. Give the probability model forXand its
parameters using the proper notation.
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d) The following events are defined:
A:X= 0 B:X= 1 C:X= 2
Implement the model in c) in StatGraphics and use it to calculate the following probabilities:
P(A) = P(B) = P(C) =
e) You explain to your friend that as well as getting the number of defective tea bags
predicted from the 95% weight limits, it is also likely that you would get a number of
defective tea bags that is either one less or one greater than this value. Event E is defined as
any one of these three outcomes occurring. You explain that the probability of E is much
greater than the probability of E not occurring. Demonstrate that probability of E not
occurring is small.
f) Considering the probability calculated in part e). Which of the following levels of
confidence could be used to express your confidence that the number of defective tea bags
selected in a sample of 20 bags will be either the number predicted from the 95% weightlimits, or one tea bag less or one tea bag more? (Underline choice(s) .)
75% 90% 95% 99%
g) You persuade your friend that it would be better to take a sample of tea bags in order to
monitor the weight, instead of measuring individual tea bags. Your friend says the limits must
be 0.032g for a 95% confidence interval for the sample mean due to the settings available on
the machine. Calculate the minimum sample size that would be required to achieve this.(Make sure that you answer is a feasible number.)