CP2403/3413: Data Analysis Under Uncertainty- Statistical Data Analysis Assignment

Internal Code: MAS794

Statistical Data Analysis Assignment:

Question: This assignment includes 6 separate analysis tasks which will give you practice in using the knowledge and techniques you are learning in the area: probability, probability distributions,sampling distributions, confidence interval estimation, hypothesis testing, regression analysis. Task 1:Relevant Topic: using Normal Distribution to determine probability or to extimate value Assume the annual mean return on ABC stock is around 15% and the annual standard deviation is around 25%. Assume the annual and daily returns on ABC stock are normally distributed. a. What is the probability that ABC will lose money during a year? b. There is a 5% chance that ABC will earn a return of at least what value during a year? c. There is a 10% chance that ABC will earn a return of less than or equal to what value during a year? d. What is the probability that ABC will earn at least 35% during a year? e. Assume there are 252 trading days in a year. What is the probability that ABC will lose money on a given day? (Hint: Let Y be the annual return on ABC and Xi be the return on ABC on day i. Then [approximately] Y = X1 + X2 + ... + X252) Task 2:Relevant Topic: using Binomial Distribution to determine probability or to estimate value Euro Watch Company assembles expensive wristwatches and then sells them to retailers throughout Europe. The watches are assembled at a plant with two assembly lines. These lines are intended to be identical, but line I uses somewhat older equipment than line 2 and is typically less reliable. Historical data have shown that each watch coming off line 1, independently of the others, is free of defects with probability 0.98. The similar probability for line 2 is 0.99. Each line produces 500 watches per hour. The production manager has asked you to answer the following questions. a. She wants to know how many defect-free watches each line is likely to produce in a given hour. Specifically, find the smallest integer k (for each line separately) such that you can be 99% sure that the line will not produce more than k defective watches in a given hour. b. Euro Watch currently has an order for 500 watches from an important customer. The company plans to fill this order by packing slightly more than 500 watches, all from line 2, and sending this package off to the customer. Obviously, Euro Watch wants to send a few watches as possible, but it wants to be 99% sure that when the customer opens the package, there are at least 500 defect-free watches. How many watches should be packed? Task 3: Relevant Topic: Using random sampling to estimate properties of a population The file “Households Survey.xlsx” contains data from a survey of 500 randomly selected households. For this problem, consider this data set a simple random sample from all possible households, where the number of households in the population is well over 1,000,000. a. Create a new variable, Total Income, which is the sum of First Income and Second Income. b. For each of the four variables Total Income, Monthly Payment, Utilities, and Debt, find the sample mean. If each of these is used as an estimate from the corresponding (unknown) population mean, is there any reason to believe that they either underestimate or overestimate the corresponding population means? Why or why not? c. What are the (approximate) standard errors of the estimates in part b? How can you interpret these standard errors? Be as specific as possible. Is the finite population correction required? Why or why not? Task 4: Relevant Topic: Simulation to understand confidence interval estimation We know that IQs are normally distributed with a mean of 100 and standard deviations of 15. Suppose you want to verify this, so you take 100 random samples of size four each and, for each sample, find a 95% confidence interval for the mean IQ. You expect that approximately 95 of these intervals will contain the true mean IQ (100) and approximately five of these intervals will not contain the true mean. Use simulation in Excel to see whether this is the case. Task 5: Relevant Topic: Using StatTools for Hypothesis Test (Paired-Sample) Eighteen readers took a speed-reading course. The file “NumberOfWords.xlsx” contains the number of words that they could read before and after the course. Test the alternative hypothesis at the 5% significance level that reading speeds have increased, on average, as a result of the course. Explain your answer in terms of the p-value. Do you need to assume that reading speeds (before and after) are normally distributed? Why or why not? Task 6: Relevant Topic: Regression analysis for estimating relationships. Firm Chair Company manufactures customized wood furniture and sells the furniture in large quantities to major furniture retailers. Imagine that you have recently been assigned to analyze the company’s pricing policy. You have been told that quantity discounts were usually given. For example, for one type of chair, the pricing changed at quantities of 200 and 400 – that is, these were the price breaks, where the marginal cost of the next chair changed. For this type of chair, the file “FirmChair.xlsx” contains the quantity and total price to the customer for 81 orders. Use regression to discover the pricing structure that Firm Chair evidently used.