homework6_320_F2023_solution

ISyE 320 Fall 2023– Homework #6 Prof. Qiaomin Xie ISyE 320 – Homework #6 Due November 2nd (Thursday) , 9PM CDT Show your work for all problems to receive full credit. Upload your scanned, handwritten solution or typed solution as a PDF to the Canvas dropbox as well as any supporting Excel spreadsheets. 1 Goodness of Fit: QQ Plots Suppose you collected 45 values representing some variate of input for your simulation project. An Excel file (hw6 data.xlsx) containing these data has been posted to Canvas with this PDF. Please use that data to perform the following exercises 1.1 Problem Create a Q-Q plot of the data against the normal distribution. Do you believe that the deviation from desired volume is a normally-distributed random variable based on your Q- Q plot? (You can use sample mean and sample standard deviation as estimates for the parameters μ and σ of a Normal distribution) Answer: See hw6 solutions.xslx for the full analysis and Q-Q plot. Based on these results, most of the data points (middle part) fall along with the 45 degree straight line, except some data points at the tails. It seems that Normal distribution with the given MLE estimators could be a good fit for the data. ♦ 2 Goodness of Fit: χ 2 -test Your team is tasked to investigate the operation efficiency of the call center for Madison Bank customer services. Your team decides to develop a simulation study. A key input component to the simulation is a model on the number of calls arriving during each minute, which is a random variable X . Your team has access to a data set that record the number of calls that have arrived during each minute of a one-hour window. The first 10 samples are as follows: 0 , 1 , 3 , 1 , 2 , 1 , 5 , 3 , 2 , 4 2.1 Problem You believe the number of calls arriving during each minute can be represented by a Poisson random variable. Based on the data, what is the value of the maximum likelihood estimator (MLE) for the parameter λ ? (You do not need to derive the formula). Page 1

ISyE 320 Fall 2023– Homework #6 Prof. Qiaomin Xie Answer: • The MLE for λ is the sample average: ˆ λ = 0 + 1 + 3 + 1 + 2 + 1 + 5 + 3 + 2 + 4 10 = 2 . 2 . ♦ 2.2 Problem Using the data above and the value of the maximum likelihood estimator ˆ λ from part (a), test the hypothesis that the random variable X follows a Poisson( ˆ λ ) distribution. Use 5 bins for your test: { 0 , 1 , 2 , 3 , ≥ 4 } for the χ 2 test with significance level α = 0 . 05 . (You can ignore the fact that the test is likely not accurate because n is too small). Answer: Recall that the Poisson PMF: P ( X = x ) = λ x e - λ x ! for x = 0 , 1 , 2 , . . . . We build the following table to compute observed frequency and expected frequency, as well as χ 2 statistics: Bin x j Observed Frequency O j Expected Frequency E j ( O j - E j ) 2 E j 0 1 10 * 2 . 2 0 e - 2 . 2 0! = 1 . 18 0.027 1 3 10 * 2 . 2 1 e - 2 . 2 1! = 2 . 44 0.128 2 2 10 * 2 . 2 2 e - 2 . 2 2! = 2 . 68 0.172 3 2 10 * 2 . 2 3 e - 2 . 2 3! = 1 . 97 0.00 ≥ 4 2 10(1 - 2 . 2 0 e - 2 . 2 0! - 2 . 2 1 e - 2 . 2 1! - 2 . 2 2 e - 2 . 2 2! - 2 . 2 3 e - 2 . 2 3! ) = 1 . 81 0.020 χ 2 0 = 0 . 347 For this test, we have k - s - 1 = 5 - 1 - 1 = 3 . We can look up the critical value in the table to get χ 2 3 , 0 . 05 = 7 . 8 . Since 0 . 347 < 7 . 8 , we fail to reject the null hypothesis that the data follow a Poisson( 2 . 2) distribution. ♦ 3 Choose your Distribution Suppose you have been observing an intersection in a remote area in northern Wisconsin. You would like to know how often semi trucks pass through the intersection. You’ve made a set of 30 observations, representing how much time (in minutes) passes between semi truck ”arrivals.” What distribution best describes this data? Answer the following questions. • Describe the data and construct a short list of distributions • Build at least two histograms Page 2