04-HW-1
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School
San Jose State University *
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Course
131
Subject
Industrial Engineering
Date
Feb 20, 2024
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docx
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Uploaded by CommodoreTankSeahorse36
ISE 131 HW #1
Due 2 PM on February 12, 2024 (Monday)
HW1.1:
Probability is Portion Based on a (Very) Large Sample – The Case of Tossing a Die:
This exercise can be considered as an extension to what was done in the class for the case
of tossing a coin. That work is summarized in the file 03-04-Coin-1-bernoulli for demonstration of necessity of huge sample size-hands-on-fill in red portion and draw graph-seed 1-completed.xlsx .
Use Excel to simulate tossing a die 10,000 times and see the slow convergence to the theoretical 1/6 occurrence proportion of any of the six possible outcomes. Through this HW exercise, you can also review some Excel functions discussed in the class and learn some new Excel functions along the way. Use the Excel template already created for you in 04-HW-1.1-tossing a die 10000 times-template.xlsx and follow the instructions there. Some hints are highlighted in green; some particular cells to pay attention to are highlighted in orange; some particular cells to pay special attention to at the end of the exercise are highlighted in red.
HW1.2:
Rediscovery of the Power of the Central Limit Theorem, particularly the central tendency, through Minitab Random Number Generation and Excel – The case of Skewed Exponential Distribution. This exercise can be considered as an extension to a demonstration of the power of the Central Limit Theorem done in the class and documented in 03-06-CLM-2-Central_Limit_Theorem-demonstration-seed 1.docx .
Follow the problem statement and the instructions given in the Excel file 04-HW-1.2-demo of central tendency of CLT-2 skewed r.v.s-template.xlsx. (It should be clear that what matters for the central tendency and for the bell shape is the SUMMING process. Dividing the sum by the number of variables involved does not matter; all it does is to shrink the distribution, in terms of the location as well as the variability, i.e., the “spread”.
(If you are curious about the speed of convergence of the distribution of the average of such skewed random variables to a Normal distribution, as the number of the random
variables increases from 2 to a large number, e.g., 30, you can repeat the process accordingly. This further investigation is optional. It takes more than four for the distribution of the average to exhibit a bell shape.)
HW1.3
: Rediscovery of the Central Tendency with Summing Two Simple Independent Random Variables – A Slightly More Complex Case Than the Case Shown in the Class.
The simple example illustrated in the class is documented in the file 03-05-illustration of the central_tendency.docx. Find and plot the probability function of X
1
and that of (
X
1
+
X
2
)/2 as what was done in that
file except that the two independent random variables of this HW exercise have this slightly more complex distribution, i
= 1 and 2:
-1 with probability 1/3
X
i
= 0 with probability 1/3
1 with probability 1/3.
HW1.4:
Use the Soft Drink data given in Exercise 5.21 on page 222 to answer the following question.
The mean and the variance of the random net weight X
are unknown. Use sample mean (i.e., the average) and sample variance and the data of Table 5E.2 to estimate the mean and the variance. Do not use the formal control chart methods yet; treat all 100 observations of Table 5E.2 as if they came from a large number of 100 observations, instead of 20 samples (or subgroups) of 5 observations each.
After you have estimated the mean and variance, use these estimates, as if you had known them all along, and the formula for the control limits for the X
chart we discussed in the class for the case of known parameters (i.e., known mean and known variance) to calculate the control limits for the X
chart of sample size 5. Use Excel only; do not use Minitab yet.
For your convenience, the data contained in Table 5E.2 have been entered into a companion Excel file entitled 04-HW-1.4&5-ex-5.21-TB5E.2-data.xlsx and has been posted on this site.
HW1.5
: Exercise 5.21 on page 222.
Part (1):
Solve this problem using Excel. Note that you do not need any Excel add-on to do the work; you can your “everyday” Excel to calculate the centerline, upper control limit and lower control limit, for each of the two control charts, and to draw the control charts themselves using the Insert Charts functions. Part (2):
In Part (1), you estimated the mean and the variance of the random thickness. Compare the estimates with the ones you produced for HW 1.4. Point out the commonalities or differences, if any. If there is any difference, which estimate should you use.
Part (3):
Solve this problem using Minitab. Look into all the option tabs to become familiar with them. Choose appropriate options so that the estimates of the mean and variance (or standard deviation) are stored on your worksheet. Also, choose appropriate options so that all the centerline, upper control limit, lower control limit and the actual plotted value for each sample (i.e., subgroup) are stored in your worksheet.
For your convenience, the data contained in Table 5E.2 have been entered into a companion Excel file entitled 04-HW-1.4&5-ex-5.21-TB5E.2-data.xlsx and has been posted on this site.
HW1.6:
Exercise 5.2 on page 221.
HW1.7:
Exercise 5.6 on page 221.
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