1. To diagnose colorectal cancer, the hemoccult test- among others- is conducted to detect occult blood in the stool. This test is used from a particular age on, but also in routine screening for early detection of colorectal cancer. Imagine you conduct a screening using the hemoccult test in a certain region. For symptom-free people over 50 years old who participate in screening using the hemoccult test, the following information is available for this region: The probability that one of these people has colorectal cancer is 0.3% (please be careful converting this to a decimal). If a person has colorectal cancer, the probability is 93.5% that the person will have a positive hemoccult test. If a person does not have colorectal cancer, the probability is 2.1% that the person will have a positive hemoccult test. a) [3] Imagine a person (over age 50, no symptoms) who has a positive hemoccult test in your screening. What is the probability that this person actually has colorectal cancer? Write the question using function notation. b) [3] If a person over age 50 years has a hemoccult test performed what is the probability that it is negative? Write the question using function notation. 2. [3] KGW News Channel 8 interviews nine Portland residents at random and asks them if they will vote for the incumbent Oregon governor this coming November. Let the random variable X measure the number of people who will vote for the incumbent governor. Write out the sample space of the random variable X. 3. [3] The random variable X is uniformly distributed, such that 15 X 42. Calculate the following P(X > 22 AND X < 35) = 4. [3] You are told that P(A) = 0.4 , P (B) = 0.3 and that the two events are independent. Calculate P(A OR B). 5. [3] You are told that events A, B are disjoint, with P(A) = 0.4 P(B) = 0.3. What is P(A and B) ? 6. Scores on the mathematics part of the SAT college entrance for a particular year derive a population mean value of 523, , with a population standard deviation, , of 112. While on the verbal part of the SAT for that same year is = 504, = 105. Let the variable X define a math score on the SAT, and let Y define a verbal score on the SAT. Furthermore define the variable T to be the total SAT score, the sum of each part of the exam; T = X + Y. a. [3] Calculate E(X +Y) b. [3] Calculate SD(X + Y) to two decimal places. 7. [3] In a drawer I have five batteries three that work and two that do not work. If I grab three batteries, what is the probability that only one works. 8. A game of chance consists of drawing one card out of ten at random. One card contains the number 100, in which a person, if they chose that card wins $100. A second card contains the number 20, if they chose that card the person wins $20. If any other card is chosen the person losses $16. The game is then played by shuffling cards, facing them down, and having the person choose a card while face down. Once the card is chosen, the outcome is determined, and the card is put back and the deck reshuffled so one can play again. Let the random variable X equal the amount of money won or lost. X $100 $20 -$16 P(X = x) 0.1 0.1 0.8 a. [3] What is the probability that a person loses two games in a row? b. [3] On average what does the person expect to win or lose in the long run per game? c. [3] Calculate , which is SD(X) to the nearest cent. 9. [3] A car dealership sells 58% of its cars to women. At that same car dealership 40% of car sales are minivans. Therefore a salesman calculates that 23.2% of the minivans are sold to women. What assumption did the salesman make to come up with this value? 10. [3] You are provided with the following information P(A) = 0.3 , P(B) = 0.2, P(A OR B) = 0.44 and that the events are independent. What is P(A | B)? Age Blood Low Pressur Normal e High Total Under 30 122 501 250 873 30 - 49 432 122 102 656 Over 50 300 205 151 656 Total 854 828 503 2185 11. A school offered blood pressure screening for its employees. The results are shown in the table. a. [4] We choose an employee at random, what is the probability that this person is a 30 to 49 year old with normal blood pressure? Write the question also in function notation along with the appropriate mathematical symbols, equal signs, to communicate your work. b. [4] If an employee is over the age of 50, what is the probability that their blood pressure is high? Write the question also in function notation along with the appropriate mathematical symbols, equal signs, to communicate your work. Blood Low Pressur Normal e High Total Age Under 30 122 501 250 873 30 - 49 432 122 102 656 Over 50 300 205 151 656 Total 854 828 503 2185 11. A school offered blood pressure screening for its employees. The results are shown in the table. c. [3] Are the events a person is Over 50, a person has high blood pressure independent events? You must show work that leads to the conclusion that yes the events are independent, or no, the events are not independent along with the statement whether the events are independent or not. Approximate to three decimal places to make your conclusion. 12. Comparison studies, such as which battery last longest or which aspirin works the best, makes one assumption as to what reality looks like. The assumption is that we have equality among the items we are comparing that is the items are equally good. What follows is the start of the model that is eventually constructed. Consider the random variable X which measures the weight of a coconut. The mean weight is 1.44 kg, and standard deviation of 0.23 kg. I randomly choose two coconuts, measure their weight individually, then subtract the weight; S = X - X. a. [3] Find the value of E(S) = b. [3] Find the value of SD(S) = 13. In a particular state university, 42% of students major in STEM fields (Science, Technology, Engineering and Mathematics). Of the students majoring in STEM fields 44% graduate in four years or less, while 52% of students in nonSTEM related fields graduate in four years or less. a. [3] What is the probability that a student chosen at random graduates in four years or less. b. [3] If a student takes more than four years to graduate, what is the probability that they majored in a STEM field? c. [3] Find P(Graduate 4 | Non-stem field) = d. [3] What is the probability that a student chosen at random is in a STEM field and takes more than four years to graduate? 14. [3] Shade in the area that corresponds to P(A OR B) 15. [3] Shade in the area that corresponds to P(NOT A AND NOT B) 16. [5] You roll a six-sided fair die. If it comes up a 6, you win $100 and that ends the play. However, if your first throw is not a 6 you get to roll again. If on the second throw you get a 6, then you win $50 and that ends the play. If you don't roll a six on the second roll you lose $25, and the play ends. Let the random variable Y equal the amount of money won or lost on one play (a play consist of rolling the die until you win or lose money). Create a probability table for all the outcomes of the random variable Y. Y P(Y = y) 17. [3] If I toss a die four times what is the probability that it lands on six all four times? 18. [3] In Iceland 56% of the population hast O type blood. If four people are chosen at random what is the probability that at least one person has O type blood in the sample of four? 19. Motor vehicles sold to individuals (non-commercial) in the U.S. are classified as either cars or light trucks (Light does not mean small, but rather that it is not a commercial truck. For example, an SUV is considered a light truck) and as either domestic or imported. In early 2004, 69% of vehicles sold were light trucks, 78% were domestic, and 55% were domestic light trucks. a. [3] Write the probability of 55% using function notation and the correct conjunction. b. [3] A vehicle is chosen at random, what is the probability that it is domestic but not a light truck? 20. [3] At a community college 28% of students work full time while attending school. Of the students working full time while attending school, 40% receive financial aid. At this same community college 59% of students receive financial aid. What is the probability that a student at this community college works full time and receives financial aid? 21. [4] Let the variable X measure the amount of liquid dispensed by a faulty soda dispensing machine at a factory. Suppose that the mean amount of soda dispensed is 2980ml, with a standard deviation of 25 ml. Furthermore, the distribution of the amount of liquid dispensed is normal. What is the probability that either the dispensing machine dispenses less than 2950 ml or more than 3000ml? Show work using function notation correctly to communicate your steps, along with conversion to a z-score. 22. Consider the random variable X which measures the weight of a coconut. The mean weight is 1.44 kg, and standard deviation of 0.23 kg. a. [3] Consider the random variable T consisting of choosing five coconuts at random, measuring each coconut weight individually and totaling the weight; T = X + X + X + X + X. Find E(T). b. [3] Consider the random variable T consisting of choosing five coconuts at random, measuring each coconut weight individually and totaling the weight; T = X + X + X + X + X. Find SD(T)