Lab_4_CLT
.pdf
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School
Texas A&M University *
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Course
350
Subject
Industrial Engineering
Date
Dec 6, 2023
Type
Pages
3
Uploaded by BailiffRiverOryx28
1
ISEN 350 Lab 4: Data Analysis, Central Limit Theorem & Sum
of Normal Random Variables
Objective:
To use Minitab software to analyze process data. To verify and
understand the central limit theorem and the sum of normally distributed
random variables through experimental simulation.
Exercise I (Minitab Data Analysis)
An audit was performed on a glass bottle supplier. A subgroup of four glass bottles was removed
from the production line and weighted every 30 minutes. A total of twenty-five subgroups were
collected. The target weight for bottle production is 250g, and engineering specifications for
bottle weight are (240g, 260g).
Data for Part I are
c
ontained in
Bottles
found on Canvas as an
Minitab worksheet
.
1)
Using Minitab, calculate the sample mean, median, mode, minimum, maximum, first and
third quartiles, range, interquartile range, sample standard deviation, skewness, kurtosis,
and sample variance for the entire data set. (
Use
Stat>Basic Statistics>Display
Descriptive Statistics)
2)
Construct a
histogram
and a
box plot
.
3)
Construct a
probability plot
for the data
4)
Construct
a run chart
for the data
Based on your results, what have you learned about bottle weight?
Does bottle weight appear
to be normally distributed?
Why or why not?
What conclusions can you draw about the ability
of the production line to produce within engineering specifications and on target? Are there any
apparent outliers in the data (boxplot will help)? If so, specify them.
Does the data appear to
have any special cause (non-random) variation affecting the glass molds? Is there any significant
evidence of clustering, mixtures, trends, or oscillations in the data (run chart)? Would you
conclude that the process operates at a
Six Sigma Quality Level (3.4 ppm defective)
? (Hand
computation may be made, or you can look at using a capability analysis, which will be covered
later in the course.)
Exercise II (Sum of Normal Random Variables)
Three shafts are made and assembled in a linkage of total length L. The length of each shaft, in centimeters,
is distributed as follows:
Shaft1:
x
1
~
N
(50, 0.040 )
Note that 0.040 is the variance, not the standard deviation.
Shaft2:
x
2
~
N
(20, 0.095)
2
Shaft3:
x
3
~
N
(36, 0.024)
(a) What is the theoretical distribution (population) of the linkage length L=X
1
+X
2
+X
3
?
Name the
distribution and define its mean and variance.
(b) Now determine the distribution of L experimentally using Minitab to simulate the production
of the linkage. Generate 3 columns of 200 rows containing random normal data, each representing
the length of the three respective shafts composing one linkage (X
1
, X
2
, X
3
). Use
Calc>Random
Data>Normal
to generate the appropriate values in each column.
Use
Calc>Row Statistics
and
select the
sum
function to find the value of L placing your linkage result in column C4.
(c) Plot a histogram for L using
Graph>histogram
, and plot a
probability plot
for L. Use
Stat>Basic Statistics>Display Descriptive Statistics
to describe the central tendency and
variation in L.
Do your applied results indicate a good fit to the normal distribution?
Does your
data agree with your theoretical results from Part (a)? What are the values for the sample mean
and standard deviation of L? Based on your experimental data, give a 95% confidence interval for
the mean of L (t-distribution)?
Exercise III. (Central Limit Theorem)
1)
A Key Process Output Variable (KPOV) is well described by a Gamma distribution with a
shape parameter of 3.0, a scale parameter of 1.0, and a threshold of 0.0.
Use Minitab to
create a worksheet containing 75 random
subgroups of size 7
(75 rows and 7 columns) from
this distribution. Use
Calc>Random Data>Gamma
to generate the required data.
Now
complete the following steps:
a)
Creating Sample Means:
-Compute the average value for each of the subgroups (rows) using
Calc>Row Statistics
and
selecting the mean function.
Store the sample means,
𝑋,
̅
in a column of your choosing.
-Generate a histogram of the sample averages,
𝑋
̅
. Note the shape of the distribution.
b)
Determining the Distribution of the Sample Mean
- Use the
Individual Distribution Identification
function in
Stat>Quality Tools
to fit the
sample mean data to a distribution. Specify fitting the normal, gamma, Weibull, and
exponential distributions. (Do not report transformations.)
-A straight-line probability plot shows the better fitting distributions. (Note the p-values).
Which distribution best fits the data?
-Do your results support the conclusion that the sample mean is normally distributed for a
sample of size 5? Explain your conclusion.
1
2
3
X
1
X
2
X
3
L
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