Mathematics

Draw on paper the solution x(t) = v1 e?1t + v2 e?2t on the x1x2-plane and then answer the questions below.(a) Based on your graph, select the correct solution curve from the interactive graph below.• Select One • Curve 1 • Curve 2 • Curve 3 • Curve 4 • None(b) Use the graph below to find lim t?+??x(t)?, where ?x(t)? is the length of the solution vector x(t).• Select One • Zero • Infinity • None(c) Introduce the unit vectors u1 = v1/?v1? and u2 = v2/?v2?. Use the graph below to find thelim t?+?u(t), where u(t) = x(t) ?x(t)?, is a unit vector in the direction of the solution vector x(t).[Select One/U1/-U1/U2/-U2/None](d) Use the graph below to find lim t????x(t)?.• Select One • Zero • Infinity • None(e) Use the graph below to find the lim t???u(t), where u(t) = x(t) ?x(t)?[Select One/U1/-U1/U2/-U2/None]9(f) Characterize the zero solution, x0 = 0.• Select One • Source Node • Source Spiral • Sink Node • Sink Spiral • Saddle • Center • NoneComments on the graph below: • The graph is interactive. • You can click on the boxes to turn on and off possible solution curves. • For each possible solution we display: the possible solution curve, the possible solution vectorx(t), and the associated unit vector u(t) = x(t)/?x(t)?. • You can move the time slider to see how each possible solution vector x(t) and unit vector u(t)change in time. • You can move the eigenvectors v1 and v2 by dragging them from the endpoint, and then see howthe curves would change.

Consider the BVP for the function y given byy??+ 25 y = 0, y(0) = 4, y (?5) = 5.(a) Find r1, r2, roots of the characteristic polynomial of the equation above.r1,r2 =(b) Find a set of real-valued fundamental solutions to the differential equation above.y1(x) =y2(x) =(c) Find all solutions y of the boundary value problem.y(x) =Note 1: If there are no solutions, type No Solution. Note 2: If there are infinitely many solutions, use k for the arbitrary constant.See in LN, § 6.1, See Examples 6.1.2-6.1.3. 2. (10 points)Consider the BVP for the function y given byy??+ 36 y = 0, y(0) = 1, y(?) = 1.(a) Find r1, r2, roots of the characteristic polynomial of the equation above.r1,r2 =(b) Find a set of real-valued fundamental solutions to the differential equation above.y1(x) =y2(x) =(c) Find all solutions y of the boundary value problem.y(x) = 1Note 1: If there are no solutions, type No Solution. Note 2: If there are infinitely many solutions, use k for the arbitrary constant.

Number of Alcoholic Drinks Consumed per Week: 5, 6, 5, 1, 7, 4, 3, 0, 3, 5, 6, 4, 2, 0, 1, 3, 8, 7, 4, 4.1. (2 points). The above distribution captures the number of alcoholic drinks, on average, twenty college age students consume in a week. Using the data, calculate the following and show your work:a. Mean:b. Median:c. Mode:d. Range:e. Variance:f. Standard deviation (note: for this and all future standard deviation calculations, please assume that we are working with a population):2. (2 points). Calculate the Z-scores for the following values randomly selected from the population: 0, 4, 5, 83. (3 points). Using the Z-scores you calculated, identify the below percentages for the area under the curve using “Appendix A – Area Under Normal Curve” and the “Calculating Z-Score Probabilities Handout”.a. What percentage of the distribution would we expect to fall between the Z-score for the value of 0 from the original distribution and a Z-score of 0?b. What percentage of the distribution would we expect to fall between the Z-score for the value of 4 from the original distribution and a Z-score of 0?c. What percentage of the distribution would we expect to fall between the Z-score for the value of 5 and the rest of the distribution to the right of that value?d. What percentage of the distribution would we expect to fall between the Z-score for the value of 0 from the original distribution and the Z-score for the value of 8 from the original distribution?

Identify the data set’s level of measurement. 1) hair color of women on a high school tennis teamA) nominal B) ordinal C) ratio D) interval 1)2) number of milligrams of tar in 85 cigarettes A) interval B) ordinal C) nominal D) ratio2)3) the ratings of a movie ranging from “poor” to “good” to “excellent” A) ratio B) nominal C) ordinal D) interval3)For questions 4 & 5: The heights (in inches) of 30 adult males are listed below.70 72 71 70 69 73 69 68 70 71 67 71 70 74 69 68 71 71 71 72 69 71 68 67 73 74 70 71 69 68Class with = ______________________________________4) Construct a frequency distribution, a relative frequency distribution, and a cumulative frequency distribution using five classes.Class Frequency MidpointRelativefrequency Cummulative frequency1Use the frequency distribution table from question #4 to answer the following question. 5) Construct a frequency polygon using five classes. Use the Midpoint for the Horizontal axis.Provide an appropriate response. 6) The table lists the smoking habits of a group of college students.Sex Non-smoker Regular Smoker Heavy Smoker Total Man 135 34 5 174 Woman 187 21 10 218 Total 322 55 15 392 If a student is chosen at random, find the probability of getting someone who is a man or a non-smoker. Round your answer to three decimal places.7) Find the probability of getting four consecutive aces when four cards are drawn without replacement from a standard deck of 52 playing cards.2Decide if the situation involves permutations, combinations, or neither. Explain your reasoning. 8) The number of 5-digit pin codes if no digit can be repeatedProvide an appropriate response. 9) How many different permutations of the letters in the word PROBABILITY are there?10) Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Find the probability of at most three boys in ten births.311)Find the area of the indicated region under the standard normal curve.Provide an appropriate response. Use the Standard Normal Table to find the probability. 12) Assume that the heights of women are normally distributed with a mean of 63.6 inches and a standarddeviation of 2.5 inches. The U.S. Army requires that the heights of women be between 58 and 80 inches. If 200 women want to enlist in the U.S. Army, how many would you expect to meet the height requirements?

Mathematics II (Applied) —BESS Assignment 3Exercise 1 Consider the square matrixA =? ? 1 2 10 1 0 1 0 1? ?(a) Calculate the eigenvalues of A and decide whether the matrix is diagonalizable. Justify your answer. (b) Find the associated eigenspaces of A. (c) Write the eigendecomposition of the power matrix A3 and of the exponential matrix eA. (d) Consider the matrixB = A ·eAProve B is diagonalizable and find the diagonal matrix D similar to B.Exercise 2 Let b : R ? R. Consider the dynamical systemx? = x2b(t)(a) Provide a suffi cient condition for the function b : R ? R such that for every initial point (x0, t0) ? R2 there exists a unique solution of the IVP. Justify your answer. (b) Puttingb(t) = etcompute the general solution ?(t;c) of the ODE. (c) Find the particular solution ?? of the IVP with starting point (x0, t0) = (0,1) and specify its domain I ? R.

A nutritionist is studying the effects of the nutrients folic acid, choline, and inositol. He has three types of food available, and each type contains the following amounts of these nutrients per ounce:Type A Type B Type CFolic acid (mg) 3 1 3Choline (mg) 4 2 4Inositol (mg) 3 2 4Suppose food type C has been improperly labeled, and it actually contains 4 mg of folic acid, 6 mg of choline, and 5 mg of inositol per ounce. Would it still be possible to use matrix inversion to find the combination of foods giving the required supply.2.5 Points

1a. What do you see as the management decision problem?1b. What is the resulting research problem?2a. Overall how satisfied are respondents with the cable service on average and in the various aspects of the Country Cable service? What informs your answer?2b. Based on your calculations, do you think the customers are satisfied with their Country Cable service? Describe the reason for your response.3. The cable company thinks it has to communicate differently to men and women and wonders if the key performance indicator (KPI) of overall satisfaction is different by gender. Articulate the null hypothesis and the alternative hypothesis Ho and Ha. What do the data say on this point? Can you accept or reject the null Ho? What statistical test informs your answer?4. If the company is concerned with weak performance on satisfaction, what do these data suggest it should focus on to improve satisfaction overall as captured in question q8. What is the null hypothesis and what is the alternative hypothesis?What aspects of satisfaction for the cable service are related to overall satisfaction (q1- q 6). Where can you accept or reject the null hypothesis? What do you think Country Cable should focus on to improve overall satisfaction? What statistical test informs your answer?5. Do you think Country Cable has addressed all their questions as a result of the ad campaign infrastructure changes and resulting survey? If you were to (re) design the study/survey, what would you do differently?

Apply elementary descriptive statistics, probability theories and quantitative methods in business in business decision making under risk and uncertainty.For the applied assignment, please submit a 5 – 6 slide PPT (NOT including title page and references page) that includes the following parts:PART 1: Create a context: Introduce a real-world business situation or an imagined business scenario that may benefit from the theories and methods presented in the prior two weeks (Weeks 3 and 4 – see the objective stated in bold above). While creating the context, ask yourself: What business use cases may benefit from the quantitative methods offered last week and this week? Pick one. You can use Google or other search tools, or your own workplace experience, to create the context. Once you decide on the context, describe it clearly in your PPT presentation. Be sure to deliver all relevant information for the audience (in this case your instructor) to help him/her understand the context fully.Part 2: Show with examples how the quantitative methodology introduced in the prior two weeks relate to the business use case you introduced. (e.g., does the math/methodology help to resolve a business problem?) Be specific in your description of how you can use the quantitative information in connection with the business case (your context) you described. Give examples. Use technical terminology when necessary.Part 3: Reflect on your learning by answering these questions: What changes have you observed in your own learning or knowledge of math/quantitative methods as a result of the topics introduced in Weeks 3 and 4 of this course? What did you find most valuable or useful for your MBA education and/or your current/future career. (As you articulate your thoughts for Part 3, be original: Do not repeat the business context or situation you described in Parts 1 and 2. Think beyond that context).

Please answer the questions that were posed in VIDEO: Chapter 1: Introduction to Social Science Research Principles and Terminology and posted below.Each of you should offer feedback to at least one of your peers.1) Criminal Justice Populations: What are 2 populations of people, places, events, or institutions that we might want to study in criminal justice research? For these different populations identify your unit of analysis.2) Think about the populations you identified, how might you use these different methodologies to collect data on these populations?3) Create a list of 5 possible variables that we might collect data on in criminal justice research.4) How would you measure the different variables you identified? For example, the age of a person could be anywhere within the number range of 0 – 100.5) Identify the appropriate scale of measurement for your variables.6) Use two of your variables to create a research statements that hypothesize relationships, identifying the dependent and independent variables.

ECON 330 Midterm Take-home Exam due Friday, April 9 before classBe sure to submit both your answers and your .R file on Courses.Name:1. What determines how much drivers are fined if they are stopped for speeding? Do demographics like age, gender, and race matter? To answer this question, we’ll investigate traffice stops and citations in Massachusetts using data from Makowsky and Stratmann (2009). Even though state law sets a formula for tickets based on how fast a person was driving, police officers in practice often deviate from the formula. The variables we use are in Table 6.12. An amount for the fine is given only for observations in which the police officer decided to assess a fine.(a) Is the effect of age on fines non-linear? Assess this question by estimating a model with a linear age term and a quadratic age term, controlling for MPHover, Female, Black, and Hispanic. Interpret the coefficients on the age variables using calculus.(b) Calculate the marginal effect of age at ages 25, 45, and 60. Calculate the age that is associated with the lowest predicted fine.(c) Do drivers from out of town and out of state get treated differently? Do state police and local police treat nonlocals differently? Estimate a model that allows us to assess whether out-of-towners and out-of-staters are treated differently and whether state police respond differently to out-of-towners and out-of-staters. To do this, add to part (a) the dummy variables OutTown and OutState, along with two interaction variables that multiply StatePol with each of the dummy variables. Interpret the coefficients on the relevant variables.(e) Test whether the two state police interaction terms are jointly significant. Briefly explain the results.2. Use globaled.csv, the data set on education and growth from Hanushek and Woessmann (2009) for this question. The variables are given in the codebook provided and Table 5.14 in the textbook.(a) Use standardized variables to assess whether the effect of test scores on economic growth is larger than the effect of years in school. At this point, simply compare the magnitude of the coefficients. We’ll do statistical tests next. The dependent variable is average annual GDP growth per year. For all parts of this exercise, control for standardized GDP per capita in 1960 and always use standardized variables in all parts.(b) Now conduct a statistical test of whether the effects of test scores and years in school on economic growth differ. Write the null hypothesis that the coefficients are the same and the appropriate alternative hypothesis. What is your conclusion about the null hypothesis?(c) Now add controls for openness of the economy and security of property rights. Which matters more: test scores or property rights? Conduct a statistical test of whether the effects of test scores and property rights differ. Write the null hypothesis that the coefficients are the same and the appropriate alternative hypothesis. What is your conclusion about the null hypothesis?