Averages, Variation, and Probability Theory

MODULE 3: AVERAGES, VARIATION, AND PROBABILITY THEORY ASSIGNMENT Questions are taken directly from Brase, Brase, Dolor, and Seibert Chapters 3 and 4, pages 126 and 179. In case an eBook page number differs, the questions are listed below: Discuss each of the following topics in class or review the topics on your own. Then, write a brief but complete essay in which you summarize the main points. Please include formulas and graphs as appropriate. Chapter 3: 1. An average is an attempt to summarize a collection of data into just one number. Discuss how the mean, median, and mode all represent averages in this context. Also, discuss the differences among these averages. Why is the mean a balance point? Why is the median a midway point? Why is the mode the most common data point? List three areas of daily life in which you think the mean, median, or mode would be the best choice to describe an average. 2. Why do we need to study the variation of a collection of data? Why isnt the average by itself adequate? We have studied three ways to measure variation. The range, the standard deviation, and, to a large extent, a box-and-whisker plot all indicate the variation within a data collection. Discuss similarities and differences among these ways to measure data variation. Why would it seem reasonable to pair the median with a box-and-whisker plot and to pair the mean with the standard deviation? What are the advantages and disadvantages of each method of describing data spread? Comment on statements such as the following: a. The range is easy to compute, but it doesnt give much information; b. although the standard deviation is more complicated to compute, it has some significant applications; c. the box-and-whisker plot is fairly easy to construct, and it gives a lot of information at a glance. 3. Why is the coefficient of variation important? What do we mean when we say that the coefficient of variation has no units? What advantage can there be in having no units? Why is relative size important? Consider robin eggs; the mean weight of a collection of robin eggs is 0.72 ounces, and the standard deviation is 0.12 ounces. Now consider elephants; the mean weight of elephants in the zoo is 6.42 tons, with a standard deviation of 1.07 tons. The units of measurement are different, and there is a great deal of difference between the weight of an elephant and that of a robins egg. Yet the coefficient of variation is about the same for both. Comment on this from the viewpoint of the size of the standard deviation relative to that of the mean. 4. What is Chebyshevs theorem? Suppose you have a friend who knows very little about statistics. Write a paragraph or two in which you describe Chebyshevs theorem for your friend. Keep the discussion as simple as possible, but be sure to get the main ideas across to your friend. Suppose they ask, What is this stuff good for? and suppose you respond (a little sarcastically) that Chebyshevs theorem applies to everything from butterflies to the orbits of the planets! Would you be correct? Explain. Chapter 4 1. Discuss the following concepts and give examples from everyday life in which you might encounter each concept. Hint: For instance, consider the experiment of arriving for class. Some possible outcomes are not arriving (that is, missing class), arriving on time, and arriving late. a. Sample space. b. Probability assignment to a sample space. In your discussion, be sure to include answers to the following questions. i. Is there more than one valid way to assign probabilities to a sample space? Explain and give an example. ii. How can probabilities be estimated by relative frequencies? How can probabilities be computed if events are equally likely? 2. Discuss the concepts of mutually exclusive events and independent events. List several examples of each type of event from everyday life. a. If AA and BB are mutually exclusive events, does it follow that AA and BB cannot be independent events? Give an example to demonstrate your answer. Hint: Discuss an election where only one person can win the election. Let AA be the event that party As candidate wins, and let BB be the event that party Bs candidate wins. Does the outcome of one event determine the outcome of the other event? Are AA and BB mutually exclusive events? b. Discuss the conditions under which P(A and B) = P(A) x P(B) is true. Under what conditions is this not true? c. Discuss the conditions under which P(A or B) = P(A) + P(B) is true. Under what conditions is this not true? 3. Although we learn a good deal about probability in this course, the main emphasis is on statistics. Write a few paragraphs in which you talk about the distinction between probability and statistics. In what types of problems would probability be the main tool? In what types of problems would statistics be the main tool? Give some examples of both types of problems. What kinds of outcomes or conclusions do we expect from each type of problem? General Instructions As doctoral students, your assignments are expected to follow the principles of high-quality scientific standards and promote knowledge and understanding in the field of criminal justice. You should apply a rigorous and critical assessment of a body of theory and empirical research, articulating what is known about the phenomenon and ways to advance research about the topic under review. Research syntheses should identify significant variables, a systematic and reproducible search strategy, and a clear framework for studies included in the larger analysis. Assignments may be written in first person (I). All assignments should be clearly and concisely written, with technical material set off. Please do not use jargon, slang, idioms, colloquialisms, or bureaucratese. Use acronyms sparingly and spell them out the first time you use them. Please do not construct acronyms from phrases you repeat frequently in the text. Structure of Assignment Paper For purposes of this assignment, there is no layout structure required as far as the setup of this paper, with one exception. I would appreciate it if you used separate headers for question 1 and question 2. Sub-headers are also allowed but not required. Questions should strive to be no less than 250 words each with no maximum limit. I expect all papers to be in the latest APA edition, properly cited, and additional resources used besides classroom textbooks. Note: Your assignment will be checked for originality via the Turnitin plagiarism tool.