Download Computer Assignment 2 Due 9/29/06 1. Solve the - RIT

Computer Assignment 2 Due 9/29/06
1. Solve the following problems using Minitab to do the calculations
A. Suppose that 10 % of all steel shafts produced by a certain process are
nonconforming but can he reworked (rather than having to be scrapped).
Consider a random sample of 500 shafts and let X denote the number
among these that are nonconforming and can be reworked.
(1 ) Find the exact probability that X
a. is at most 55?
MTB > cdf 55;
SUBC> binomial 500 .1.
Cumulative Distribution Function
Binomial with n = 500 and p = 0.1
x
55
P( X <= x )
0.795771
b.less than 55?
MTB > cdf 54;
SUBC> binomial 500 .1.
Cumulative Distribution Function
Binomial with n = 500 and p = 0.1
x
54
P( X <= x )
0.752307
c. between 45 and 60 inclusive.
MTB >
SUBC>
MTB >
SUBC>
MTB >
MTB >
cdf 60 c1;
binomial 500 .1.
cdf 44 c2;
binomial 500 .1.
let c3=c1-c2
print c3
Data Display
C3
0.730212
MTB
(2 ) Do the problem again this time using the normal approximation to the
binomial distribution with the continuity correction.
MTB > cdf 55.5;
SUBC> normal 50 6.708.
Cumulative Distribution Function
Normal with mean = 50 and standard deviation = 6.708
x
55.5
P( X <= x )
0.793868
MTB > cdf 54.5;
SUBC> normal 50 6.708.
Cumulative Distribution Function
Normal with mean = 50 and standard deviation = 6.708
x
54.5
MTB >
SUBC>
MTB >
SUBC>
MTB >
MTB >
P( X <= x )
0.748839
cdf 60.5 c4;
normal 50 6.708.
cdf 44.5 c5;
normal 50 6.708.
let c6=c4-c5
print c6
Data Display
C6
0.735111
MTB >
3) Do the problem again this time using the normal approximation to the
binomial distribution without the continuity correction.
Which of (2) and (3) give a better approximation to the exact value?
MTB > cdf 55;
SUBC> normal 50 6.708.
Cumulative Distribution Function
Normal with mean = 50 and standard deviation = 6.708
x
55
P( X <= x )
0.771979
MTB > cdf 54;
SUBC> normal 50 6.708.
Cumulative Distribution Function
Normal with mean = 50 and standard deviation = 6.708
x
54
P( X <= x )
0.724514
MTB >
SUBC>
MTB >
SUBC>
MTB >
MTB >
cdf 60 c7;
normal 50 6.708.
cdf 45 c8;
normal 50 6.708.
let c9=c7-c8
print c9
Data Display
C9
0.703966
Using the continuity correction gives a better approximation.
B. Suppose that only . .1 % of all computers of a certain type experience
CPU failure during the warranty period. Consider a sample of 30,000
computers.
(1 ) What is the expected value and standard deviation of the number of
computers in the sample that have the defect ? (Do this part by hand.)
µ = 30000(.001) = 30
! = 30000(.001)(.999) = 5.474
or
! = 30 = 5.477
(2 ) Using an appropriate binomial distribution find the probability that
more than 15 sampled computers have the defect ?
MTB >
SUBC>
MTB >
MTB >
cdf 14 c10;
binomial 30000 .001.
let c11=1-c10
print c11
Data Display
C11
0.999083
(3 ) Using an appropriate Poisson distribution approximate the probability
that more than 15 sampled computers have the defect ?
MTB > cdf 15 c12;
SUBC> poison 30.
MTB > let c13=1-c12
MTB > print c13
Data Display
C13
0.998053
(
4) Comment on the goodness of the approximation.
The answers tp (2) and (3) are very close as would be expected.
2. Use Minitab to do this problem that demonstrates The Central Limit
Theorem
Put the numbers 0, 1,2,3,4 into column 1.
Put the first five digits of your Social Security Number in c2.
Enable the command language.
At the prompt put in the command
let c3 = c2/sum(c2)
Now do the following computations and show your results on the page
stapled to the front of the computer output.
1. Verify that the numbers in c3 are a probability distribution.
2. Find the mean of the probability distribution in c3.
3. Find the standard deviation of the probability distribution in c3.
You will now simulate your distribution 100 times.
calc>randomdata>discrete
In dialog box
Generate 100 rows of data
Store in columns c5-c104
Values in c1
Probabilities in c3
OK
enter the commands
rmeans c5-c104 c105
sort c105 c106
print c106
Make a stem and leaf display for c50 using the pull down menu or the
command
stem-and-leaf c50
describe c50
Make a stem and leaf display for c106 using the pull down menu or the
command
stem-and-leaf c106
describe c106
Make normal probability plots for c50 and c106.
Answer the following questions on the sheet stapled to the front of your
computer output.
1. Based on the stem and leaf display and the normal probability plot for
c50 does the data appear to be normally distributed? Explain your
answer.
2. Answer the same question for c106.
3. What is the mean and the standard deviation obtained in the describe
command for c50 and c106?
4. What should the mean and standard deviation be in theory for c50? for
c 106?
5. Compare the mean and standard Deviation in questions 3 and 4 by
finding the percentage error?
6. State the Central Limit Theorem carefully and explain how the results
you obtain in c106 validate it for your problem.
A typical answer to this problem is available at
http://www.rit.edu/~mjgsma/smam314winter03/ca2sol.pdf
The solution will be different for students who work independently but
the conclusions should be similar. If you get a large percentage error for
your mean and standard deviation check your calculation of the
theoretical mean and standard deviation. Chances are pretty good that
you made a mistake.