Assignment 2 - Statistics 160 - Fall 2024 PDF

STATISTICS 160 - Fall 2024 Assignment 2 Due: 11:59pm on October 4th, 2024 Note: Assignment Student Solutions must be uploaded on URCourses as a single file in PDF format. 1. A group of 100 students are applying for s...

STATISTICS 160 - Fall 2024 Assignment 2 Due: 11:59pm on October 4th, 2024 Note: Assignment Student Solutions must be uploaded on URCourses as a single file in PDF format. 1. A group of 100 students are applying for scholarships to enter one of three universities. University A offers a scholarship of $100000 to one student, University B offers a scholarship of $50000 to one student and University C offers a scholarship of $10000 to one student. It is assumed that students cannot receive more than one scholarship. (a) How many ways can the three universities assign their scholarships to the stu- dents? (b) Suppose Shrek and Fiona are two students out of the 100 students applying for these scholarships. What is the probability that Shrek and Fiona both receive a scholarship of any amount? (c) What is the probability that Shrek and Fiona both receive a scholarship where Fiona receives a higher amount? (d) Suppose someone found out that Shrek was going to receive a scholarship. Given this new information, what is the probabiltiy that Fiona will receive a scholarship? 2. McVittie’s ice cream shop is a store that sells people random combinations of in- gredients in an ice cream sundae to add some excitement/fear in the ordering pro- cess. McVittie’s sundaes consist of three ingredients: Ice Cream, Fudge and Topping. McVittie offers 5 flavours of ice cream (Strawberry, Chocolate, Vanilla, Pistacchio, Rocky Road), 3 types of fudge (Dark Chocolate, White Chocolate, Cookie Dough) and 4 types of topping (Peanuts, Bananas, Cherries, Sprinkles). It is assumed that a sundae can only contain one type of ice cream, one type of fudge and one type of topping. (a) How many types of sundaes are offered by McVittie’s ice cream shop? (b) What is the probability that your sundae will contain peanuts as a topping? (c) What is the probability that your sundae will contain either dark chocolate or white chocolate fudge? (d) What is the probability that you sundae will not contain chocolate ice cream? 1 STATISTICS 160 - Fall 2024 3. Thirty percent of those sent to jail are repeat offenders. Twenty percent of those sent to jail have been convicted of a violent crime. Ten percent are repeat offenders and have been convicted of a violent crime. (a) What is the probability that someone who is sent to jail has neither been convicted of a violent crime nor is a repeat offender? (b) If it is known that an individual is a repeat offender, what is the probability that they were convicted of a violent crime? (c) Are the events of being a repeat offender and convicted of a violent crime inde- pendent? 4. Let X be a discrete random variable with the probability distribution given below: x -1 2 6 9 p(x) 0.1 0.2 0.3 0.4 (a) Calculate µ = E(X). (b) Calculate E(X 2 ). (c) Verify that σ 2 = E((X − µ)2 ) = E(X 2 ) − µ2 by calculating both sides of the equation and showing they are equal. 5. A new surgical procedure is successful with a probability of p. Assume that the oper- ation is performed five times and the results are independent of one another. What is the probability that (a) all five operations are successful if p = 0.80? (b) exactly four operations are successful if p = 0.60? (c) less than two operations are successful if p = 0.30? 2