MODULE 4: THE BINOMIAL PROBABILITY DISTRIBUTION, NORMAL CURVES, AND SAMPLING DISTRIBUTIONS ASSIGNMENT Questions are taken directly from Brase, Brase, Dolor, and Seibert Chapters 5 and 6, pages 225 and 301, respectively. In case an eBook page number differs, the questions are listed below: Chapter 5: 1. Discuss what we mean by a binomial experiment. As you can see, a binomial process or binomial experiment involves a lot of assumptions! For example, all the trials are supposed to be independent and repeated under identical conditions. Is this always true? Can we always be completely certain that the probability of success does not change from one trial to the next? In the real world, there is almost nothing we can be absolutely sure about, so the theoretical assumptions of the binomial probability distribution often will not be completely satisfied. Does that mean we cannot use the binomial distribution to solve practical problems? Looking at this chapter, the answer seems to be that we can indeed use the binomial distribution even if not all the assumptions are exactly met. We find in practice that the conclusions are sufficiently accurate for our intended application. List three applications of the binomial distribution for which you think, although some of the assumptions are not exactly met, there is still adequate reason to apply the binomial distribution. 2. Why do we need to learn the formula for the binomial probability distribution? Using the formula repeatedly can be very tedious. To cut down on tedious calculations, most people will use a binomial table such as the one found in Appendix II of this book. a. However, there are many applications for which a table in the back of any book is not adequate. For instance, compute P(r = 3) where n = 5 and p = 0.735. Can you find the result in the table? Do the calculation by using the formula. List some other situations in which a table might not be adequate to solve a particular binomial distribution problem. b. The formula itself also has limitations. For instance, consider the difficulty of computing P(r285) where n = 500 and p. 0.6. What are some of the difficulties you run into? Consider the calculation of P(r = 285). You will be raising c. 0.6 and 0.4 to very high powers; this will give you very, very small numbers. Then, you need to compute C500,285, which is a very, very large number. When combining extremely large and extremely small numbers in the same calculation, most accuracy is lost unless you carry a huge number of significant digits. If this isnt tedious enough, consider the steps you need to compute P(r285) = P(r = 285) + P(r = 286) + … + P(r285)? Does it seem clear that we need a better way to estimate P(r285)? In Chapter 6, you will learn a much better way to estimate binomial probabilities when the number of trials is large. 3. In Chapter 3, we learned about means and standard deviations. In Section 5.1, we learned that probability distributions also can have a mean and standard deviation. Discuss what is meant by the expected value and standard deviation of a binomial distribution. How does this relate back to the material we learned in Chapter 3 and Section 5.1? 4. In Chapter 2, we looked at the shapes of distributions. Review the concepts of skewness and symmetry, then categorize the following distributions as to skewness or symmetry: a. A binomial distribution with n = 11 trials and p = 0.50. b. A binomial distribution with n = 11 trials and p = 0.10. c. A binomial distribution with n = 11 trials and p = 0.90. In general, does it seem true that binomial probability distributions in which the probability of success is close to 0 are skewed right, whereas those with probability of success close to 1 are skewed left? Chapter 6: 1. If you look up the word empirical in a dictionary, you will find that it means relying on experiment and observation rather than on theory. Discuss the empirical rule in this context. The empirical rule certainly applies to the normal distribution, but does it also apply to a wide variety of other distributions that are not exactly (theoretically) normal? Discuss the terms mound-shaped and symmetric. Draw several sketches of distributions that are mound-shaped and symmetric. Draw sketches of distributions that are not mound-shaped or symmetric. To which distributions will the empirical rule apply? 2. Why are standard z values so important? Is it true that z values have no units of measurement? Why would this be desirable for comparing data sets with different units of measurement? How can we assess differences in quality or performance by simply comparing z values under a standard normal curve? Examine the formula for computing standard z values. Notice that it involves both the mean and the standard deviation. Recall that in Chapter 3, we commented that the mean of a data collection is not entirely adequate to describe the data; you need the standard deviation as well. Discuss this topic again in light of what you now know about normal distributions and standard z values. 3. Most companies that manufacture a product have a division responsible for quality control or quality assurance. The purpose of the quality-control division is to make reasonably certain that the products manufactured are up to company standards. Write a brief essay in which you describe how the statistics you have learned so far could be applied to an industrial application (such as control charts and the Antlers Lodge example). 4. Most people would agree that increased information should give better predictions. Discuss how sampling distributions actually enable better predictions by providing more information. Examine Theorem 6.1 again. Suppose that x is a random variable with a normal distribution. Then x the sample mean based on random samples of size n, also will have a normal distribution for any value of n = 1, 2, 3… What happens to the standard deviation of the x distribution as n (the sample size) increases? Consider the following table for different values of n. Due to space, see the book to complete this question. 5. In a way, the central limit theorem can be thought of as a kind of grand central station. It is a connecting hub or center for a great deal of statistical work. We will use it extensively in Chapters 7, 8, and 9. Put in a very elementary way, the central limit theorem states that as the sample size n increases, the distribution of the sample mean x will always approach a normal distribution, no matter where the original x variable came from. For most people, it is the complete generality of the central limit theorem that is so awe-inspiring: It applies to practically everything. List and discuss at least three variables from everyday life for which you expect the variable x itself not to follow a normal or bell-shaped distribution. Then, discuss what would happen to the sampling distribution x if the sample size were increased. Sketch diagrams of the x distributions as the sample size n increases. 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 all tables attached. Note: Your assignment will be checked for originality via the Turnitin plagiarism tool.