Assignment 2 Statistics

Part I. Consists of 10 written questions. Q-1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 marks A business owner places a product in both the front and back of his store. Let F be the event of selling well when a product is placed at the front of the store. Let B be the event of selling well when a product is placed at the back of the store. A product is selected at random, and the following probabilities are given: P(F ∩ B) = 0.25, P(F ∩ BC) = 0.15, P(F C ∩ B) = 0.10, and P(F C ∩ BC) = 0.50. (a) (2 pts) What is the probability that the randomly selected product sells well at the front of the store? (b) (1 pt) If the randomly selected product sells well at the front of the store, what is the probability it will also sell well at the back of the store? (c) (1 pt) If the randomly selected product sells well at the back of the store, what is the probability it will also sell well at the front of the store? Q-2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 marks Statistics Canada wanted to find the distribution of ages of Indigenous men living in Canada who are employed. The researcher drew a random sample of some 300 Indigenous men from the government records and found the following distribution: Age Groups Total Employment Status < 25 years old (A1) 25-40 years old (A2) > 40 years old (A3) Employed (E) 25 59 36 120 Unemployed (U) 72 54 54 180 Total 97 113 90 300 One family was selected at random from government records of families with Indigenous men. What is the probability that the randomly selected Indigenous man is either employed or over 40 years of age? Q-3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 marks A farmer in Alberta has 1000 cows. 400 of these cows like to roam free in the pastures (event A); 100 of them were born on the farm (event B); and 300 of them are solely grain-fed (event C). We assume that events A, B and C are independent. What is the probability of finding a cow which meets all three of these criteria? Q-4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 marks A beauty salon sells spa packages in which you can choose from one of two nail treatments, one of three facial treatments, and one of four massage treatments. How many different packages are available? Q-5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 marks When students register for online classes, they have 2 options: (a) a basic, free class (b) a premium class with added resources and tools costing $12/month. Based on this year’s student data, the school finds that 70% of new students choose the basic, free class. At the end of their first month, students are faced with 3 options: (i) the basic, free class, (ii) the premium class, or (iii) dropping the class altogether. The past year’s student data also shows that the proportions of students with the basic, free class in their first month who choose these 3 options are (i) 60% (ii) 30% (iii) 10%, respectively. Also, the proportions of students with the premium class in their first month who choose these 3 options are (i) 80% (ii) 10% (iii) 10%. (a) (5 pts) Use a tree diagram to find the probability of a student dropping the class at the end of their first month? Define all relevant events and write any probabilities in terms of these events. In solving any part of this question, explicitly write down the formulas and any steps you use. (b) (2 pts) Is dropping the class in the 2nd month independent of having the free class in the 1st month? c 2021 A.Almaskut, W.Horn, F.Said Page 2 of 4 Next page. . . Stat 2507 Introduction to Stat Modelling I Due: Monday, Feb 22, 2021 (c) (4 pts) What is the probability that a student had originally registered to the free class in their first month given that he/she has the premium class in their second month? (d) (4 pts) What is the probability that a student has a premium class in the first month or a premium class in the second month? Q-6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 marks On a given day, of all surveyed commuters in Ottawa, 60% used public transit, 90% used their own vehicle, and 55% used both public transit and their own vehicle. Define events T and O as follows: • Let T denote the event a commuter who used public transit • Let O denote the event a commuter who used own vehicle (a) (2 pts) What is the probability that a randomly selected commuter uses either their own vehicle, or public transit, or both? (b) (3 pts) What is the probability that a randomly selected commuter uses public transit or their own vehicle, but not both? (c) (2 pts) What is the probability that a randomly selected commuter uses neither public transit nor their own vehicle? Q-7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 marks A sample of skin care products is selected from one of two skin care lines: Yves Saint Laurent (S1) and Clinique (S2), with probabilities P(S1) = 0.80 and P(S2) = 0.20. If the products have been selected from S1, the probability of observing a positive change in your skin, i.e. event A, is P(A|S1) = 0.15. Similarly, if the products have been selected from S2, the probability of observing A is P(A|S2) = 0.25. In solving any part of this question, explicitly write down the formulas and any steps you use. (a) (4 pts) If a sample is randomly selected and event A is observed, what is the probability that the sample was selected from population S1? (b) (4 pts) If a sample is randomly selected and event A is observed, what is the probability that the sample was selected from population S2? c 2021 A.Almaskut, W.Horn, F.Said Page 3 of 4 Next page. . . Stat 2507 Introduction to Stat Modelling I Due: Monday, Feb 22, 2021 Part II. SPSS Questions. Please include your output. Q-8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 marks When applying for jobs, assume each job application has a 30% chance of resulting in an interview independent of each other job application. Use the relative frequencies to estimate the probability of being called for an interview. This requires applying to a number of jobs and computing the relative frequency for the number of times you are called for an interview. Using SPSS, simulate the experiment of applying to jobs n times. Let I denote the event of being called for an interview. Consider n = 10, 100, 5, 000 and 10, 000 simulations. (a) (2 pts) Use the relative frequency method to estimate P(I) for n = 10. (b) (2 pts) Use the relative frequency method to estimate P(I) for n = 100. (c) (2 pts) Use the relative frequency method to estimate P(I) for n = 5, 000. (d) (2 pts) Use the relative frequency method to estimate P(I) for n = 10, 000. (e) (2 pts) Compare your results from these simulations and identify which of the point estimates is closest to the true probability. Explain

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