Advanced Statics - open to bidding

Please find the document attached. To be completed by tomorrow noon. 15.1 Suppose Hans flips a coin 100 times, getting 45 heads. You thought he should get 50 heads. Hans’ data are Bernoulli, y1, y2, …, y100, where each yi value is either 0 or 1 and where Σyi = 45. Answer the question, “Is the difference from 45 to 50 in Hans’ data explainable by chance alone?” as follows: A. State the relevant null model for the data Y1, Y2, …, Y100. B. Find the approximate null distribution of the test statistic T = ΣYi using the central limit theorem. Use your model in Exercise 15.1A to find the relevant mean and variance. C. Use the distribution of T that you found in Exercise 15.1B to find the two-sided p-value for Hans’ data, and then answer the question. 15.2 Ten adult dogs on a diet of Standard Dog Chow are observed for a day using video surveillance. The total length of time Xi, out of 24 hours, that the dogs appeared to be smiling, was recorded for each of dogs i = 1, 2, …, 10. The next day, the dogs were all fed Happy Dog Chow. The video surveillance was repeated, and the total length of time Yi, out of 24 hours, that the dogs appeared to be smiling, was recorded for each dog. The researchers then calculated the differences, Di = Yi − Xi, for each of dogs i = 1, 2, …, 10, getting data 0.22, −1.79, 1.17, −1.46, 1.36, 1.20, 0.60, 0.62, −0.75, and −0.28. A null model is that the Di are iid with E(Di) = 0. A. Find an approximate null distribution of the test statistic D Di = ∑ (1/10) . B. Use the distribution in Exercise 15.2B to find the p-value for the two-sided test. Are results explainable by chance alone? C. You can create a bootstrap population distribution where the null is true by subtracting d di = ∑ (1/10) from each of the data values di. Using this bootstrap population distribution produce the histogram of 1000 values of the test statistic and locate your observed test statistic on the graph. D. Using the bootstrap data in Exercise 15.2C find the two-sided p-value and compare your results with Exercise 15.2B. 15.3 BankTen conducted a survey of 1250 customers’ satisfaction on a 1, 2, 3, 4, 5 scale, with 1 being “Highly Dissatisfied” and 5 being “Highly Satisfied.” The average response was a 3.05, with a standard deviation of 1.4. Further breakdown of the data into groups of n1 = 250 loyal customers versus n2 = 1000 newer customers shows averages 3.50 and 2.94, respectively. Answer the question, “Is the difference between these averages explainable by chance alone?” as follows: A. State the iid null model. Explain the meaning of each unknown parameter of this model. (There are five unknown parameters and they add to 1.) B. Find the approximate distribution of Y ‾1 − Y ‾2, assuming the iid null model of Exercise 15.3A is true. C. Using the distribution in Exercise 15.3B find the p-value for the two-sided test and state your conclusion. D. The approximation in Exercise 15.3B is suspect. Use a simulation model that produces iid Likert scale data to estimate the probability in Exercise 15.3C. Comment on whether the approximation in Exercise 15.3B is reasonable.