What are the requirements? Consider external sources if you need to research to help define your problem. What are the ethical implications of a project like this? How can solving the problem be helpful to people or contribute to society in some way?

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Implementation & Improvement/Plan & Prototype

Use these instructions and questions when writing your lab reports to ensure you cover all necessary parts of Engineering Design and the DMAIC process.

Define/Ask & Research – Determine the problem or the goal of your work. What are you trying to develop or solve? Do you have a specific goal to attain? Who is it for, and how will it be presented?

What are the requirements? Consider external sources if you need to research to help define your problem. What are the ethical implications of a project like this? How can solving the problem be helpful to people or contribute to society in some way?

Measure – Engineering design usually involves either be given access to data or have some information regarding what you are measuring so you can collect your own data. What data is either available or being collected? How is this data being used? What is the frequency of data collection? How are you using this data? If measuring your own data, be sure to discuss how it is being collected and from where.

Analyze/Imagine – Based on the problem and information available, consider potential solutions for your problem. This may sometimes involve teammates, depending on the assignment or project being completed. Try to be creative and build on other ideas if needed. This section is key for ensuring good teamwork! Ensure everyone’s ideas are included and have a good conversation about potential solutions.

Create a design based on your selected solution.

This does not necessarily mean your design will work right away. Create a prototype and test it. If it does not work, consider how you can redesign your solution to better meet your needs. You may go through multiple iterations before reaching a final stage.

Control/Test – How are you ensuring success of your solution? Can your solution be repeated? Keep a record of any testing done, as well as the frequency/times of testing, to ensure that your results are both repeatable and reliable.

Once you have addressed each of these sections, be sure to include some form of conclusion to explain how the overall project went for you/your team. What did you learn? How could the lab be improved? Do you understand the concept driving this lab?

Identify the feasible region; and Plot the objective function lines and find the optimal solution. What is the optimal solution? What is the optimal value of the objective function?

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PROBABILITY

Directions:

Submit a paper copy of your solution on November 22nd (Tuesday) in class. You may use graph paper to solve the optimization problem using the graphical method.

Remember to sign the honors pledge as shown below.

Problem 1 (15 points)

A manufacturing firm produces two products. Each product must undergo an assembly process and a finishing process. It is then transferred to the warehouse, which has space for only a limited number of items. The firm has 80 hours available for assembly and 120 hours for finishing, and it can store a maximum of 10 units in the warehouse. Each unit of Product 1 has a profit of $50 and requires 4 hours to assemble and 12 hours to finish. Each unit of Product 2 has a profit of $70 and requires 10 hours to assemble and 8 hours to finish. The firm wants to determine the quantity of each product to produce in order to maximize profit.

Solve the linear optimization problem using the graphical method:

Plot the constraints;

Identify the feasible region; and

Plot the objective function lines and find the optimal solution.

What is the optimal solution? What is the optimal value of the objective function?

 

Analyze a population, perhaps using actual birth, survival, and death rates, using the eigenvalue approach to dynamical systems.

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Final Project Description

 

OVERVIEW

The application project should demonstrate your understanding of a particular problem from another discipline, the mathematical concepts and techniques that can be used to solve the problem, and the ways in which mathematical modes of thought are brought to bear on the problem. The project should be an extension of mathematical ideas or applications seen in this course.

 

REQUIREMENTS FOR THE PROPOSAL

Due November 23

Describe a particular problem from another discipline that can be addressed through the use of one or more of the linear algebra techniques was have studied this semester.

You should:

pose one or more specific and sufficiently interesting questions relevant to your problem and make the case that these techniques could help someone answer those questions.

not attempt to use these techniques to actually solve the problem you describe in this part of the project (yet).

complete the statement: “In this project I/we will…”

convince me that you have the resources to complete your project.

do more than argue that linear algebra can be applied to a certain area (e.g. physics, engineering, computer graphics, etc.)

You are welcome to explore an interesting problem that has already been solved (see below for suggestions where to look for one of these) or pose an original problem that has not yet been solved. You should include any data relevant to the problem you choose—or instructions for how someone could obtain such data.

The proposal should be submitted in one of the following two formats:

a one page paper, 12 point Times New Roman or similar font, double-spaced with proper formatting, references (required – and cite in text if and only if you list in the bibliography) and citations, OR

a face-to-face (possibly on Zoom) presentation.

The proposal should be given as if to a fellow student in Math 3305. Thus you may assume that your audience is familiar with the material we have covered together as a class this semester. I will provide you with feedback on your proposal, discussing the feasibility of your proposed project and suggesting ideas for enhancing your project, specifically about references, linear algebra methods, creativity and clarity of main question, and feasibility of the project.

 

CITING YOUR SOURCES

You must cite your sources appropriately in text or in your slides, and a list of references must appear at the end of your paper or presentation, formatted appropriately. It does not matter which formatting style you use (APA, MLA, etc.), but you should be consistent in your formatting. Be sure to format your citations (footnotes, endnotes, etc.) correctly according to the formatting style you choose. (Just including a footnote with a link to a Web page is not sufficient.) Make clear what references you use and how you use them. I will look up all of your references so make it clear for me how to find them. If you include a website url include the whole thing. Did most of your paper come from a single journal article? If so, then there is no need to cite it in every paragraph, but you should explain your use of the article clearly in your paper and cite direct quotations from the article. At most 20% of your paper can be cited from other work; you need to write the majority of the content yourself. Do not cut and paste paragraphs or computations from your sources into your paper. You cannot cite examples from other papers; create your own examples to demonstrate your understanding.

Did your paper come from a variety of sources? Make it clear what ideas (facts, computations, analysis, etc.) came from which sources through proper citations. That is the spirit of not plagiarizing—making it clear where you obtained your ideas. Whatever else you do, do not plagiarize! If your life is falling apart and you are tempted to plagiarize to save time or get a good grade, please talk to me instead. I would rather grant you an extension than send you before the Honor Council for plagiarism—but I will send you to the Honor Council if it comes to that. Papers and slides will be required to be submitted via Safe Assign which is an automatic check for plagiarism.

Also, pay attention to the quality of any sources you reference. Determine if you have a reputable source or not. Wikipedia is not an acceptable source—not because its entries are user-generated, but because no encyclopedia is an acceptable source. You’re welcome to use Wikipedia as a starting point for your research, however. Most statements in Wikipedia are referenced at the bottom of the Web page. Follow these references for sources to use instead of Wikipedia itself. Another student’s project is also not an acceptable source. Again, it could give some ideas of where to start only. Use an in text citation if and only if the work being cited appears in your bibliography. Do not list sources in the bibliography without specifying in the text of the paper how you used them. In general your paper should cite at least one textbook or other scholarly or professionally reviewed source.

 

TOPIC IDEAS (You should not re-use a project done in any other class.) Note that you must apply in some way a topic that has been introduced in the class post-exam 2: coordinate systems, eigenvalues, diagonalization, etc.

  1. Markov Chains – Model a board game, sport, or real-world situation using a Markov chain, then determine optimal strategies through an analysis of the model. Or use the basic ideas behind Google’s PageRank algorithm for ranking the popularity of Web pages to rank some other set of connected items. If you choose this topic you may need to model a very simplified version due to computational constraints. These can also be applied to biology, for example, model inheritance of genetic traits
  1. Discrete Dynamical Systems – Analyze a population, perhaps using actual birth, survival, and death rates, using the eigenvalue approach to dynamical systems (this is covered later in our textbook in Chapter 5—you can read ahead if you are interested in this). Use the Leslie model to solve a problem in ecology; e.g. look at Hawaii’s population and analyze it with actual birth, survival and death rates

 

  1. Social networks analysis using graphs and adjacency matrices.

 

  1. Obtaining a closed formula for the Fibonacci Sequence. (This and other “pure math” applications are acceptable.)

 

  1. Explain the proof or application of the Jordan Canonical Form of a matrix. This is a deep theorem of interest to those in pure mathematics, as well as a very useful formula that appears in a variety of applied mathematics scenarios.

 

  1. Choose one of the applied mathematics projects from the Boyce Applied Mathematics textbook. (Knowledge of differential equations required!) Or look at 5.7 in our book about applications of differential equations.

 

  1. Use real-life season data to create a win-loss differential matrix and obtain the rankings of sports teams.

 

  1. Section 5.6 : Discrete Dynamical Systems and predatory-prey models

 

  1. Learn how to tell when a linear transformation is 1-1 and onto. Show that any finite dimensional vector space is isomorphic to ℝ n  using the coordinate map.

 

  1. Section 5.4: Eigenvectors and Linear Transformations.

 

  1. Section 7.4: The Singular Value Decomposition and applications to machine learning.

 

  1. Finding the nth power or a square matrix; finding the exponential of a matrix

 

Some additional topics students did present in past semesters are: using linear algebra to create new Sudoku puzzles from old; Linear algebra applications in electrical engineering: analyzing electric circuits; the cubic formula in linear algebra; Markov chains to model music playlists; how to take the exponential of a matrix; Merlin’s Magic Squares; Singular Value Decomposition; using affine transformations to create fractals, modeling the game Chutes and Ladders; modeling populations using census data.

 

One place to find topics is in scholarly journal articles in engineering or other journals. Many such journal articles are available online as PDFs through the Library.  Presenting some of the content of such an article (perhaps with some supporting content from one or two other sources) in a clear manner, understandable to a fellow Math 3305 student would be an appropriate way to approach this application project.

 

 

Develop a scatter plot showing the variables, rate of service and worker hours, with the y variable on the vertical axis and the x variable on the horizontal axis, and indicating the type of relationship.

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Linear Regression Analysis

Terry Downes owns a commercial cleaning company. He has conducted a survey of customers to determine how satisfied they are with the work performed. He devised a 100-point rating scale—with 0 being poor and 100 being excellent service, selected a random sample of 14 customers, and asked them to rate the service. He also recorded the number of worker hours spent in the customer’s facility.
These are in the file named Downes (see attached).

Use Excel to complete the following:

  • Develop a scatter plot showing the variables, rate of service and worker hours, with the y variable on the vertical axis and the x variable on the horizontal axis, and indicating the type of relationship.
  • Develop a linear regression model to explain the variation in the service rating.

Use Word to complete the following:

  • Describe the model, showing the results of the pertinent hypothesis tests, using a significance level of 0.10.

Submit your work in a Word document and attach your Excel file.

Calculate the impact of the energy score program using a “simple difference.” What is the counterfactual in this case? Calculate the impact of the energy score program using the “before and after” method. What is the counterfactual in this case?

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AEC 434 Problem Set 3

Total Points: 30
1. A researcher is interested in understanding whether or not home energy scores, as suggested recently in Corvallis, will make newly built homes more energy efficient. Imagine that this researcher is able to conduct an experiment on selected cities in Oregon.
This researcher randomly selects treatment and control newly-built homes. “Treatment” means that the home is given an energy score. “Control” homes do not get an energy score.
The research measures average energy efficiency of newly built homes before the experiment and two years after the experiment. The average energy efficiency measures for new homes (where higher is more energy efficient), before and after the experiment are below:

a. Calculate the impact of the energy score program using a “simple difference.” What is the counterfactual in this case?

Hint: The counterfactual is the outcome you are comparing against (i.e. what you use as a baseline). 2 pts

b. Calculate the impact of the energy score program using the “before and after” method. What is the counterfactual in this case? 2 pts

c. Calculate the impact of the permit program using “difference in differences” (or double- difference). 2 pts
Group Energy efficiency before
experiment
Energy efficiency after
experiment
Homes with scores 30 60
Homes without scores 40 42

2. Lincoln County has hired your economic consulting firm to calculate the net benefits from recreation visits to Seal Rock Wayside. Admission to Seal Rock is currently free. You conduct some field work and initial research that yields the following information:

City/Town Miles from
Seal Rock
Population
(1000 s)
Number of
Vehicles
Estimated
visitors
Visit rate
(per 1000
people)
Corvallis 58 58 35 250 4.3
Albany 70 54 15 130 2.4
Newport 9 11 40 1,000 90.9
Lincoln
City
32 10 30 700 70.0
Yachats 14 1 25 200 200.0

You estimate that each vehicle carried 3.5 persons (2.2 adults), on average, and make the following assumptions:
– The average operating cost of vehicles is $0.65 per mile.
– The average speed on county highways is 50 miles per hour.
– The opportunity cost to adults of travel time is 30 percent of their wage rate; it is zero for children.
– Adult visitors have the average county wage rate of $12 per hour.
This information is summarized in the attached Excel spreadsheet.
a. Calculate travel cost per vehicle and travel cost per person for each town. 3 pts.

b. Create a scatter plot with travel cost per person on the X axis and visit rate on the Y axis.
Fit a line through the data and report the slope. Make sure to include your plot with your answer. 3 pts

c. You know demand when admission is free. Find additional points on the demand curve by predicting visitors from each town as the fee goes up in $10 increments, until the demand is zero. Follow these steps:
– Calculate new rate = old rate + (fee*slope)
– New number of visitors = New rate * Population
– Get total number of visitors and plot D curve 4 pts

d. Estimate annual benefit to visitors by calculating the area under the demand curve. 3 pts

3. In an example for use of the Value of a Statistical Life in BCA we calculated the benefits of increasing the speed limit for a highway (Week 8). To get additional practice and a better understanding of the use of VSL, this exercise asks you to calculate the benefits of reducing the speed limit. Suppose the area around a 30-mile stretch of highway has become more urbanized and traffic has increased, so city planners are considering lowering the speed limit on the highway as it passes the town from 65 mph to 50 mph. The relevant information is available in the attached Excel spreadsheet. Use the worked example from week 8 as a guide to answer the following questions.

a. Calculate additional travel time costs by calculating average travel time before and after, then calculating the difference. Then calculate total time added. 2 pts
b. Calculate costs as the value of time added to leisure and business travelers. 2 pts
c. Calculate benefits as avoided operating, fatal crash, and non-fatal crash costs. 2 pts
d. Calculate NSB. Should the speed limit be lowered? 2 pts
4. An economist has conducted a survey of ranchers to find out their willingness to pay for adopting rotational grazing practices that benefit sage grouse populations. The survey yielded the following data:
Stated price (annual payment, $) Fraction of respondents accepting stated price
(%)
0 97
15 92
30 86
45 79
60 66
75 50
90 35
105 22
120 11
Calculate the mean Willingness to Pay (WTP) for the sampled population.

Hint: Use equation (16.1) from the text. 3 pts.

What is the value of the account, in dollars, after 30 years? What is the exact instantaneous rate of change of the value of the account at exactly 9 years? Give your answer rounded to two decimal places

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CALCULUS

When an initial amount of money, A,A, in dollars, is invested into an account that earns interest continuously, the Future Value of the account after tt years is given by the formula: F(t)=Aert,F(t)=Aert, where rr is the annual interest rate earned by the account. Let A=$24,000A=$24,000 and r=7.9%r=7.9%.

  1. A) What is the value of the account, in dollars, after 30 years? Give your answer rounded to two decimal places.

Answer $

  1. B) What is the exact instantaneous rate of change of the value of the account at exactly 9 years? Give your answer rounded to two decimal places.

Answer: dollars per year

  1. C) At what time, in years, is the instantaneous rate of change of the value of the account increasing by $37,412.18 per year? If necessary, round your answer to two decimal places.

Answer: After years.

  1. D) What is the average rate of change of the future value of the account between year 30 and year 33 (i.e. slope of the secant line connecting the points)? (Round to the nearest penny/cent.)

Answer: dollars per year. (Round to two decimal places.)

 

Create a table of descriptive statistics for this variable. Create a “Normal Q-Q plot” for this variable. Create a histogram for the “EATDRINK” variable.

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SPSS Homework: Testing Data for Normality Assignment Instructions

Overview

This assignment is designed to increase your statistical literacy and proficiency in summarizing, visualizing, and interpreting variability within data sets. You will work within SPSS, using data that are related to specific research scenarios in the behavioral sciences such as psychology, social work, and counseling. A working knowledge of how to summarize and visualize data is fundamental to being able to understand and apply research in your related field to your practice. Additionally, SPSS skills are professionally valuable, as it is one of the most used statistical software packages in behavioral science settings, both academic and professional.

Instructions

  • Download the file titled SPSS Homework Data File. It is an SPSS data file that contains a selection of data from the American Time Use Survey. This survey is conducted by the US Bureau of Labor Statistics and collects information on the amount of time people report spending on different daily activities, from childcare to work to leisure. This file contains data from the year 2019.
  • The file contains the following variables:
    • SLEEP: Time spent sleeping (minutes per day)
    • EATDRINK: Time spent eating and drinking (minutes per day)
  • You will use the information in this file to respond to a series of questions asking you to create and interpret various tables and charts in SPSS. Paste your SPSS output directly into the homework underneath each relevant question. Type written answers into the document underneath the question to which they relate.
  • This assignment is worth 60 points.

Review the Watch: SPSS Homework tutorial: Testing Data for Normality and Creating Z Scores in this module for directions on how to enter data in SPSS and use it to produce the final product for your assignment. The scenarios begin on the next page.

 

Problem Set 1:

  1. For the “EATDRINK” variable, using the “Explore” command, test the data for normality (as shown in the SPSS tutorial) by performing the following steps:
  2. Create a table of descriptive statistics for this variable. Paste the table here: (6 pts)
  3. Run a Kolomogorov-Smirnov/ Shapiro-Wilks test for this variable. Paste the table here: (6 pts)
  4. Create a “Normal Q-Q plot” for this variable. Paste the table here: (6 pts)
  5. Create a histogram for the “EATDRINK” variable. Paste the histogram here: (6 pts)
  6. In several sentences, discuss whether the daily minutes of eating and drinking people report is normally distributed. Justify your answer using information from numbers 1 and 2 above. (6 pts)

 

Problem Set 2:

  1. For the “SLEEP” variable, using the “Explore” command, test the data for normality by performing the following steps:
  2. Create a table of descriptive statistics for this variable. Paste the table here: (6 pts)
  3. Run a Kolomogorov-Smirnov/ Shapiro-Wilks test for this variable. Paste the table here: (6 pts)
  4. Create a “Normal Q-Q plot” for this variable. Paste the table here: (6 pts)
  5. Create a histogram for the “SLEEP” variable. Paste the histogram here: (6 pts)
  6. Some might argue that the amount of sleep people report is normally distributed. Based on careful evaluation of ALL of the data in numbers 4 and 5, would you agree or disagree that minutes of “Sleep” is normally distributed in this sample? Support your answer with information from the data and the reading/presentations this week. (6 pts)

 

 

Find the values of c to make the following fimetions valid probe- Nay density functions.

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Probability

Question *a. Let X be a Bernoulli random variable with P(X = 0) = I — p, P(X = I)= p. Let Y = I — X and Z = XY. (a) (5 points) Fuld P(X = x, Y = y) for x, y, e (0, (b) points) Find P(X = x,Z = z) for x, y,a a {0, 2. Question so. X and Y are independent Poisson random variables with perm- mess a and

  • (a) (5 points) Show that X+ Y has a Poisson distribution with parameter a +ft
  • (b) (5 points) Show that the conditional distribution of X, given X, Y = n is binomial and fmd its parameMts.

Question 23. Find the values of c to make the following fimetions valid probe- Nay density functions.

  • (a) (5 Paints) f(x)= cfgl —x)}-, for 0< x< I (Hint the derivative of sin-‘(x)= v.,=)
  • (b) (5 points) f(x). ceop(—x — for x e R
  • (5 Points) f(x)= C(5.x’ — coa(x)) for 00 x51

Question X is a uniform random variable distributed on [0, ‘xJ. Find the pdf of Y = sinX.

Question .5. Xis a continuous random variable with pdf, x >0

fx(x) = 0, x 5 0

  • (a) (4 Paints) Find EIX1
  • (b) (3 points) Find EIX1 (c) (3 points) Find Var[X]

 

What is SIMD, OMP, and MPI? What are the differences between them? Describe what is shown on the graph you have produced.

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Distributed parallel computing

Goal of this assignment

  • SIMD using vectorized data
  • Symmetric Multiprocessing using OpenMP
  • Distributed Memory using Message Passing Interface (MPI)

Problem Statement Given the following algorithm doing matrix multiplication, implement multiples variations using different types of parallel processing we saw in class: SIMD, OMP, MPI, and OMP+MPI.

for (i = 0; i < N; i++)

for (j = 0; j < N; j++) {

c[i][j] = 0; for (k = 0; k < N; k++)

c[i][j] += a[i][k] * b[k][j];}

Step 1

Run Matrix Multiplication non-vectorized in C

Create a vectorized SIMD matrix multiplication version in C

Run HelloWorld.c MPI

Lab Parallel Computing – On

Read input matrix from two files as described in MPI and OpenMP Approaches to consider .docx

Download MPI and OpenMP Approaches to consider .docx in section: main program. This will be used for demo and grading.

Matrix multiplication in C on Wolfgand cluster with OpenMP.

Matrix multiplication in C on Wolfgand cluster with MPI (Distributed Memory)

Update graph to include SIMD, OpenMP and MPI versions. (You can removed unoptimized algorithm as it expected will be “off the chart” and make the chart difficult to read)

Extra credit: Matrix multiplication in C on Wolfgand cluster with both OpenMP and MPI.

Tasks expected to be completed:

  • Automate running matrix multiplication on different size matrix and generating data in tabular format for graph production.
  • Matrix multiplication in C on Wolfgand cluster without SIMD and without parallelization.
  • Matrix multiplication in C on Wolfgand cluster with SIMD non-vectorized (w/o -O3) and SIMD vectorized (w/ -O3) (rewrite algo accordingly and try with and without -O3).
  • Produce a single graph comparing the speed of all implementations
  • Writing

Research Question:

  • What is SIMD, OMP, and MPI? What are the differences between them?
  • Describe what is shown on the graph you have produced.

 

What is the value of the account, in dollars, after 13 years? hat is the exact instantaneous rate of change of the value of the account at exactly 8 years?

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Calculus

When an initial amount of money, A,A, in dollars, is invested into an account that earns interest continuously, the Future Value of the account after tt years is given by the formula: F(t)=Aert,F(t)=Aert, where rr is the annual interest rate earned by the account. Let A=$15,000A=$15,000 and r=7%r=7%.

 

  1. A) What is the value of the account, in dollars, after 13 years? Give your answer rounded to two decimal places.

 

Answer $

 

  1. B) What is the exact instantaneous rate of change of the value of the account at exactly 8 years? Give your answer rounded to two decimal places.

 

Answer: dollars per year

 

  1. C) At what time, in years, is the instantaneous rate of change of the value of the account increasing by $7,197.91 per year? If necessary, round your answer to two decimal places.

 

Answer: After years.

 

  1. D) What is the average rate of change of the future value of the account between year 13 and year 15 (i.e. slope of the secant line connecting the points)? (Round to the nearest penny/cent.)

 

Answer: dollars per year. (Round to two decimal places.)