Download Stats Workshop 3

MATH10002 Introduction to Statistics - Workshop 3 –
Calculating Probabilities
A large part of this exercise makes use of the additive properties of Normal random variables which are
given in detail in chapter 4 of the notes for this module. These are extremely important in statistics and the
following questions are designed to get you to think about and apply these ideas to a practical problem.
Please make sure that you show your postgraduate student your solutions at the end of the session in order
to get the appropriate credit.
You might be best doing the arithmetic calculations on a pocket calculator or the Windows desktop
calculator on the PC but they can also be done in Minitab.. Make sure that you switch on “Enable
Commands” in the session window. Minitab can be used to calculate probabilities associated with a
random variable, X. You need to choose Calc > Probability Distributions from the menus and then click
on the required distribution. You will mainly be doing cumulative probability calculations (note that for a
discrete distribution it gives the probability that the random variable is less than or equal to the number x)
but question 5 requires you to use the inverse cumulative probability facility.
The core of a transformer consists of 50 layers of sheet metal with 49 insulating paper layers. The
thickness of a layer of a single metal sheet is Normally distributed with mean 0.5mm and standard
deviation 0.05mm. The thickness of an insulating paper layer is Normally distributed with mean 0.05mm
and standard deviation 0.02mm. the core of the transformer, once produced, should fit neatly into a box of
height 28mm.
1.
A random sample of 50 metal sheets is taken from the manufacturing process. The company feels
that as long as the total thickness of the 50 sheets is less than 26mm the process is working
correctly. What is the probability that the sample results will indicate that the process is not
working correctly.
2.
Let W be the total thickness of the combined layers of sheet metal and insulating paper that make
up the core of the transformer. What proportion of the transformers will be rejected because they
do not fit their boxes?
3.
What is the exact probability, that in a day’s production of 50 transformers, less than 3
transformers will not fit their boxes? Calculate the same probability using an approximating
distribution (use the correction factor) and compare your two answers answers by calculating the
error in your approximation relative to the true value.
4.
What is the exact probability, that in a day’s production of 100 transformers, less than 5
transformers will not fit their boxes? Calculate the same probability using an approximating
distribution (use the correction factor) and compare your two answers by means of the relative
error in your approximation. Do we have a better approximation here than in part 3, and if so
why?
5.
What should the height, h, of the box be if the proportion of transformers fitting their boxes is to
be 0.96?
6.
If the height of the boxes used to house the transformers is in fact Normally distributed with mean
28mm and standard deviation 0.2mm what proportion of the transformers will be rejected because
they do not fit their boxes?
7.
The company now decides to change the way it produces the transformers. It proposes to produce
large sheets of insulating paper. A large sheet is cut up into 49 smaller sheets which are used in
the same transformer. The thickness of a large sheet can be controlled to be constant throughout
but there is a variability in thickness from sheet to sheet such that the thickness of a large sheet has
the Normal distribution with mean 0.05mm and standard deviation 0.02mm. What is now the
distribution of the total thickness W of the combined layers of sheet metal and insulating paper
that make up the core of a transformer? What proportion of transformers made in this way will be
rejected by the company because they do not fit their boxes, assuming again that boxes are exactly
of height 28mm? Is the new proposal an improvement or not on the original scheme?