Reference no: EM133253716

Assignment - Statistics Questions

Question 1 - (100-200 words) (Please have 2-3 references)

Explain Bayes' theorem, accompanied by an example.

Question 2 - (400-500 words) (Please have 2-3 references)

In the book Making Hard Decisions: An Introduction to Decision Analysis, Robert T. Clemen presents an example in which he discusses the 1982 John Hinckley trial. In describing the case, Clemen says: In 1982 John Hinckley was on trial, accused of having attempted to kill President Reagan. During Hinckley's trial, Dr. Daniel R. Weinberger told the court that when individuals diagnosed as schizophrenics were given computerized axial tomography (CAT) scans, the scans showed brain atrophy in 30% of the cases compared with only 2% of the scans done on normal people. Hinckley's defense attorney wanted to introduce as evidence Hinckley's CAT scan, which showed brain atrophy. The defense argued that the presence of atrophy strengthened the case that Hinckley suffered from mental illness.

1. Approximately 1.5 percent of the people in the United States suffer from schizophrenia. If we consider the prior probability of schizophrenia to be .015, use the information given to find the probability that a person has schizophrenia given that a person's CAT scan shows brain atrophy.

2. It can be argued that .015 is not a reasonable prior probability of schizophrenia. This is because .015 is the probability that a randomly selected U.S. citizen has schizophrenia. However, John Hinckley was not a randomly selected U.S. citizen. Rather, he was accused of attempting to assassinate the President. Therefore, it might be reasonable to assess a higher prior probability of schizophrenia. Suppose you are a juror who believes there is only a 10 percent chance that Hinckley suffers from schizophrenia. Using .10 as the prior probability of schizophrenia, find the probability that a person has schizophrenia given that a person's CAT scan shows brain atrophy.

3. If you are a juror with a prior probability of .10 that John Hinckley suffers from schizophrenia and given your answer to part 3, does the fact that Hinckley's CAT scan showed brain atrophy help the case that Hinckley suffered from mental illness?

4. If you are a juror with a prior probability of .25 that Hinckley suffers from schizophrenia, find the probability of schizophrenia given that Hinckley's CAT scan showed brain atrophy. In this situation, how strong is the case that Hinckley suffered from mental illness?

Question 3 - (400-500 words) (Please have 2-3 references)

What makes a function of a discrete variable a candidate for a discrete random variable distribution? What about the counterpart of this candidacy in the case of a continuous variable?

Respond to these questions by visually communicating your responses in an infographic.

Tips on Designing an Infographic

Build an infographic and attach a written summary for items which cannot fit on the infographic.

An information graphic (infographic) is a visual representation of a data set or instructive material. Infographics take a large amount of information in text (or numerical form) and then condense it into a combination of images and text highlights. This modern data transformation technique allows viewers to quickly grasp essential insights about a specific subject. Review this ten-minute video (link below) to see what elements go into creating an effective infographic. It will provide some background on how to complete your assignment for this week.

How to Create an Infographic - Part 1: What Makes a Good Infographic?

Get as creative as possible, and design a well-organized, easy to understand infographic. You can visit Piktochart, Canva, Venngage, and Visme online for infographic examples, tips on how to create them, and templates.

Share your completed infographic in GAP. Be sure to share the file in a format your instructor can view such as PDF. Search the help section of the tool you chose to use to find out the options for downloading, sharing, and publishing your infographic.

Piktochart Download Options

Canva Download Support

Venngage Download Support

Visme Download Support

Question 4 - (400-500 words) (Please have 2-3 references)

In an article in the Journal of Retailing, Kumar, Kerwin, and Pereira noted factors affecting merger and acquisition activity in retailing. As part of the study, the authors compared the characteristics of "target firms" (firms targeted for acquisition) and "bidder firms" (firms attempting to make acquisitions). Among the variables studied in the comparison were earnings per share, debt-to-equity ratio, growth rate of sales, market share, and extent of diversification.

1. Let be the mean growth rate of sales for all target firms (firms that have been targeted for acquisition in the last five years and that have not bid on other firms), and assume growth rates are approximately normally distributed. Furthermore, suppose a random sample of 25 target firms yields a sample mean sales growth rate of 0.16 with a standard deviation of 0.12. Use critical values and this sample information to test H0: ? = .10 versus Ha: ? >.10 by setting ? equal to .10, .05, .01, and .001. How much evidence is there that the mean growth rate of sales for target firms exceeds .10 (that is, exceeds 10 percent)?

2. Now let ? be the mean growth rate of sales for all firms that are bidders (firms that have bid to acquire at least one other firm in the last five years), and again assume growth rates are approximately normally distributed. Furthermore, suppose a random sample of 25 bidders yields a sample mean sales growth rate of 0.12 with a standard deviation of 0.09. Use critical values and this sample information to test H0: ? = .10 versus Ha: ? > .10 by setting ? equal to .10, .05, .01, and .001. How much evidence is there that the mean growth rate of sales for bidders exceeds .10 (that is, exceeds 10 percent)?

Question 5 - Neurons Activity (1-2 Page) (Please have 2-3 references)

Figure 1 shows a typical composition of neurons in a net. This discussion questions aimed at learning how natural composition of neural nets can be mimicked to derive algorithms that can provide prediction at higher levels of complexity.

1. Consider a multiple regression model of your choice containing two predictors. Calculate the values of the regression equation with a range chosen by you for the values of the predictors. Provide the graph of the regression equation which is a plane.

2. Augment your regression model by a sigmoid posterior filter. Calculate the values of the regression equation within the range chosen in part 1. Provide the graph of the regression equation, which will be a sigmoidal surface. Show that this graph cannot have a local extremum.

3. Now, add another regression model with sigmoid posterior containing the same two predictors in the previous part. Repeat parts 1 and 2 for this new regression model.

4. Now consider a model of a neural net, which has two parallel hidden layers, which are the two regression models considered above. The output of the net is simply the superposition of the two regression models. Show that by proper choice of the parameters you can have an output, which has a local extremum within the choice for the range of the values of the predictor. Provide the graph of the regression equation.

5. How does this observation indicate an application of neural networks modeling in practice?