Assignment 3-Part 2

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University of Ottawa *

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2303

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Industrial Engineering

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Dec 6, 2023

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pdf

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3

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Assignment 3 - Part 2 ADM2303, Fall 2023 • The assignment is due on Friday, Nov 17 at 11:59 pm • The total point for this assignment is 34 points • For information regarding penalties, refer to the course outline "Part 2 Marking Scheme" • The solution file must be type-written and submitted on Brightspace in PDF format • You can use Microsoft Excel to carry out your calculations. However, you must show the detailed step by step calculations and process in your solution file. • Include a statement of academic integrity in your submission. • You are responsible for completing this assignment on your own. The integrity statement prohibits receiving assistance in answering questions through any form of service. Question 1. Ear temperature and hypothermia (8 marks) According to a 1992 study published in the Journal of the American Medical Association, the average oral body temperature is 98.2°F. It's interesting to note that body temperatures measured through the ear tend to be approximately 0.5°F higher than those taken orally. For the purpose of this exercise, let's make this adjustment and consider a normal distribution model for ear temperatures. The model has an average (mean) ear temperature of 98.7°F and a standard deviation of 0.7°F. a) Hypothermia, a medical emergency characterized by an abnormally low body temperature, may be indicated by an ear temperature of 97°F or lower. Based on our normal distribution model, what percentage of people are at risk of having ear temperatures that may suggest hypothermia? (2 marks) b) Calculate the interquartile range (IQR) for ear temperatures using the given information. The IQR gives us an idea of the variability around the average temperature. (2 marks) c) A new company has developed an ear thermometer that claims to be more accurate than existing models. While the average ear temperature reading remains unchanged at 98.7°F, this new thermometer boasts an IQR of just 0.5°F. Calculate the standard deviation of ear temperatures as measured by this new thermometer. (2 marks) d) It's flu season, and a temperature of 99.5°F or higher may be indicative of a fever. What is the probability that a randomly selected individual will have a fever based on our original normal distribution model? (2 marks) Question 2. Tropical City Pool Homes ROI Analysis (10 marks) Sunshine Pools Inc. installs swimming pools across a tropical city and is interested in estimating how much of the pool's cost homeowners can recoup when selling their homes. Consider a homeowner who installs a pool for $30,000 and sells the home for $20,000 more than they would have without the pool. In this case, the homeowner has recouped 20,000 30,000 × 100 of the pool's cost. To investigate, the company randomly selects 80 homes sold in the tropical city over the past 6 months. All homes are between 2,000 and 3,500 square feet. Variables include:
• Selling price (in $1,000s) • Square footage • Number of bathrooms • Niceness rating (1 to 7, assigned by the real estate agent) • Presence of a pool (1 for yes, 0 for no) Data for this study is found in the accompanying file, "Pool_Home_Data.xlsx." a) Calculate the following descriptive statistics for homes with and without pools: Count, Mean, Sample Variance, Sample Standard Deviation, Minimum, Maximum, Range. (2 marks) b) Using the statistics calculated in Part A, compare the mean selling prices of homes with and without pools. (1 mark) c) Create a boxplot to visually compare the distribution of selling prices for homes with and without pools. Describe any patterns or outliers you observe from the boxplot. (2 marks) d) Calculate the 25th, 50th (median), and 75th percentiles for the selling prices of homes with and without pools. How do these values compare, and what do they tell you about the distribution of selling prices for each group? (2 marks) e) Assume that the average cost of installing a pool in the sampled homes is $32,500. Estimate how much of this cost could be recouped when selling the home. (2 marks) f) Discuss why the comparison made in Part B might be misleading. Consider variations in square footage, number of bathrooms, and niceness ratings among the homes. (1 mark) Question 3. Machine Breakdown Analysis (8 marks) The maintenance department of a factory claims that the frequency of breakdowns for a specific machine follows a Poisson distribution. According to them, the machine experiences an average of two breakdowns every 500 hours. Let X be an exponential random variable representing the time (in hours) between successive breakdowns. a) Calculate the λ parameter, which represents the rate of breakdowns per hour. Also, find μ X , the average time (in hours) between successive breakdowns. (1 mark) b) Formulate the Probability Density Function f(x) for the exponential distribution of X, using the parameters identified in Part a. (1 mark) c) Use software like Excel to plot the Probability Density Function curve for X. Display your results here. (2 marks) d) Assuming the maintenance department's claim is accurate, determine the probability that the time between successive breakdowns is at most 5 hours. (1 marks) e) Still assuming the maintenance department's claim is valid, calculate the probability that the time between successive breakdowns falls between 100 and 300 hours. (2 marks)
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