Tutorial 1 Questions

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University of Toronto Department of Mechanical and Industrial Engineering MIE236: Probability (Fall 2023) Tutorial 1 Problem 1: If ? and ? are two sets then ? ∩ (? ∪ ?) 𝑐 is equal to : A. B B. A C. ∅ D. ? ∪ ? Problem 2: Consider the following Venn diagram, where ?, ? and ? are three sets. Let the number of elements in ? be denoted by 𝑛(?) . which is equal to 90. What is the value of 𝑛(?) + 𝑛(?) + 𝑛(?) − 𝑛(? ∩ ?) − 𝑛(? ∩ ?) − 𝑛(? ∩ ?) + 𝑛(? ∩ ? ∩ Z)? A. a+b+43 B. a+b+63 C. a+b+96 D. a+b+106 Problem 3: In a city, there are 3 major newspapers A, B and C of which at least two are read by 35% of population. A and B are read by 15%. C is read by 45% and that all the three are read by 10%. Then the percentage of people who read the newspaper C alone is A. 20 B. 10 C. 15 D. 25 E. 5

University of Toronto Department of Mechanical and Industrial Engineering MIE236: Probability (Fall 2023) Problem 4: A study of 2007 model automobiles was conducted. In the study the following variables were considered: the Region in which the car was manufactured (Europe, North America, Asia); the Type of automobile (compact, midsize, large), volume of the engine in liters, and the type of Fuel used (regular, premium, 85% Ethanol). The variables Region, Type, volume, and Fuel are, respectively are: A. quantitative, categorical, categorical, quantitative. B. categorical, quantitative, quantitative, categorical. C. categorical, categorical, quantitative, quantitative. D. categorical, categorical, quantitative, categorical. E. Unable to determine without knowing the values of the various variables. Problem 5: The boxplot of the ages (in years) of the employees in a company is shown below. Read the following statements regarding the distribution age of these employees: I. The distribution is skewed to the right. II. The interquartile range is 8 years. III. The median is about 40 years. Which of the above three statements is (are) correct? A. Only III is true. B. Only II and III are true. C. Only I and II are true. D. Only I is true. ! E. Only II is true.

University of Toronto Department of Mechanical and Industrial Engineering MIE236: Probability (Fall 2023) Problem 6: For data on a quantitative variable, which of the following is true? A. If the distribution is symmetric, then the range is an appropriate measure of the center. B. If the distribution is symmetric, then the standard deviation is an appropriate measure of the center. C. If the distribution is skewed, then the mean is an appropriate measure of the center. D. If the distribution is skewed, then the median is an appropriate measure of the center. E. If the distribution is skewed, then the standard deviation is an appropriate measure of the spread. Problem 7: Of the following measures: mean, median, IQR (inter-quartile range), and standard deviation, which measures are resistant to outliers? A. Mean and median B. Median and IQR C. Mean and standard deviation D. Median and standard deviation E. None of the above Problem 8: A magazine commissioned a study to find the number of days in the standard high- school year in different countries and provinces. A boxplot is shown below. Which is the correct statement to compare the mean and the median? A. They should be similar. B. Mean is larger than the median. C. Can't decide based on this information. D. Median is larger than the mean.

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University of Toronto Department of Mechanical and Industrial Engineering MIE236: Probability (Fall 2023) Problem 9: The blood urea nitrogen count (BUN) is a measure of how well the kidney is working (a lower count is better). 20 patients had their blood urea nitrogen count measured, with the results shown in the histogram below. Note that the BUN is a whole number. A value that falls exactly on a class boundary (like 12) is placed in the bar to the right. How many of the patients have a BUN count between 12 and 18 (including 12 but not including 18)? What is the proportion that the patients have a BUN count between 12 and 18 among all patients? A. 50% B. 55% C. 60% D. 65% E. 70%

University of Toronto Department of Mechanical and Industrial Engineering MIE236: Probability (Fall 2023) Problem 10: The boxplot below is of the birthweights (in ounces) of 188 infants born in a local hospital. How many infants have birthweights between 103 and 117 ounces? Note: In order to avoid unnecessary complications, you may assume that no infant had a birthweight exactly equal to 103 or 117 ounces. A. 47 B. 141 C. 113 D. 94 E. None of the other given choices gives the correct answer. Problem 11: A population of 10 Real Analysis students obtained the following scores on an exam: Student 1 2 3 4 5 6 7 8 9 10 Score 82 62 80 58 72 73 65 66 74 62 The mean score is 69.4 and the standard deviation is 8.04 . If we collect 100 simple random samples of size 3 from this population, which of the following statements are TRUE? I) The average of the 100 sample means will be approximately 69.4. II) The histogram of the sample means would be approximately normal III) The standard deviation will be approximately 8.04

University of Toronto Department of Mechanical and Industrial Engineering MIE236: Probability (Fall 2023) A. Only one statement is true B. II) and III) C. All of the statements are true D. I) and III) E. I) and II) F. No Answer

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