trabajo_entregar_2012

8/2/2019 Trabajo_entregar_2012

1/6

Page 1 of 6

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.)


2/6

Page 2 of 6

a) Calculate the capacity of the process.


3/6

Page 3 of 6

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:


4/6

Page 4 of 6

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.


5/6

Page 5 of 6

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.


6/6

Page 6 of 6

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.)

trabajo_entregar_2012

Documents