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Module 5 Test Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculate the specified probability 1) Suppose that T is a random variable. Given that P(-3.1 ≤ T ≤ 3.1) = 0.25, and that P(K < -3.1) = P(K > 3.1), find P(K < -3.1). A) 0.25 B) 1.55 C) 0.375 D) 0.75 Answer: C Objective: (5.1) Find Probability Using Rules Determine the possible values of the random variable. 2) The following table displays a frequency distribution for the number of siblings for students in one middle school. For a randomly selected student in the school, let X denote the number of siblings of the student. What are the possible values of the random variable X? Number of siblings 0 1 2 3 4 5 6 7 Frequency 189 245 102 42 24 13 5 2 A) 0, 1, 2, 3, 4, 5, 6, 7 C) 189, 245, 102, 42, 24, 13, 5, 2 B) 7 D) Brother, sister Answer: A Objective: (5.1) Find Possible Values of Random Variable Find the specified probability. 3) A statistics professor has office hours from 9:00 am to 10:00 am each day. The number of students waiting to see the professor is a random variable, X, with the distribution shown in the table. x 0 1 2 3 4 5 P(X = x) 0.05 0.10 0.40 0.25 0.15 0.05 The professor gives each student 10 minutes. Determine the probability that a student arriving just after 9:00 am will have to wait at least 10 minutes to see the professor. A) 0.95 B) 0.40 C) 0.20 D) 0.10 Answer: A Objective: (5.1) Determine Probability of Event Use random-variable notation to represent the event. 4) Suppose that two balanced dice are rolled. Let X denote the absolute value of the difference of the two numbers. Use random-variable notation to represent the event that the absolute value of the difference of the two numbers is 2. A) {X = 2} B) {(1, 3), (2, 4), (3, 5), (4, 6), (3, 1), (4, 2), (5, 3), (6, 4)} C) X = 2 D) P{X = 2} Answer: A Objective: (5.1) Use Random-Variable Notation to Represent Event 1 Provide an appropriate response. 5) True or false? For any discrete random variable, the possible values of the random variable form a finite set of numbers. A) True B) False Answer: B Objective: (5.1) *Know Concepts: Discrete Random Variables Obtain the probability distribution of the random variable. 6) When a coin is tossed four times, sixteen equally likely outcomes are possible as shown below: HHHH HHHT HHTH HHTT HTHH HTHT HTTH HTTT THHH THHT THTH THTT TTHH TTHT TTTH TTTT Let X denote the total number of tails obtained in the four tosses. Find the probability distribution of the random variable X. Leave your probabilities in fraction form. A) B) C) D) x P(X = x) x P(X = x) x P(X = x) x P(X = x) 0 1/16 0 1/16 0 1/16 1 1/4 1 1/8 1 3/16 1 1/4 2 7/16 2 3/8 2 1/2 2 3/8 3 1/4 3 1/8 3 3/16 3 1/4 4 1/16 4 1/16 4 1/16 4 1/16 Answer: C Objective: (5.1) Obtain Probability Distribution of Random Variable Calculate the specified probability 7) Suppose that K is a random variable. Given that P(-2.85 ≤ K ≤ 2.85) = 0.175, and that P(K < -2.85) = P(K > 2.85), find P(K > 2.85). A) 0.825 B) 0.4125 C) 1.425 D) 0.175 Answer: B Objective: (5.1) Find Probability Using Rules 8) Suppose that W is a random variable. Given that P(W ≤ 2) = 0.425, find P(W > 2). A) 2 B) 0.575 C) 0.425 D) 0 Answer: B Objective: (5.1) Find Probability Using Rules Use random-variable notation to represent the event. 9) Suppose that two balanced dice are rolled. Let X denote the sum of the two numbers. Use random-variable notation to represent the event that the sum of the two numbers is less than 4. A) {X < 4} B) (1, 1), (1, 2), (2, 1) C) {X+Y < 4} D) {X ≤ 4} Answer: A Objective: (5.1) Use Random-Variable Notation to Represent Event 2 Determine the possible values of the random variable. 10) Suppose that two balanced dice, a red die and a green die, are rolled. Let Y denote the value of G - R where G represents the number on the green die and R represents the number on the red die. What are the possible values of the random variable Y? A) -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 B) 0, 1, 2, 3, 4, 5, 6 C) 0, 1, 2, 3, 4, 5 D) -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 Answer: A Objective: (5.1) Find Possible Values of Random Variable 11) For a randomly selected student in a particular high school, let Y denote the number of living grandparents of the student. What are the possible values of the random variable Y? A) 0, 1, 2 B) 4 C) 0, 1, 2, 3, 4 D) 1, 2, 3, 4 Answer: C Objective: (5.1) Find Possible Values of Random Variable Find the mean of the random variable. 12) The random variable X is the number of golf balls ordered by customers at a pro shop. Its probability distribution is given in the table. x 3 6 9 12 15 P(X = x) 0.14 0.13 0.36 0.27 0.10 A) 8.34 B) 9.18 C) 6.51 D) 9 Answer: B Objective: (5.2) Find Mean of Random Variable Given Probability Distribution Find the expected value of the random variable. 13) Sue Anne owns a medium-sized business. Use the probability distribution below, where X describes the number of employees who call in sick on a given day. Number of Employees Sick P(X = x) 0 0.1 1 0.35 2 0.3 3 4 0.2 0.05 What is the expected value of the number of employees calling in sick on any given day? A) 2.00 B) 1.75 C) 1.85 D) 1.00 Answer: B Objective: (5.2) Find Expected Value of Discrete Random Variable Find the mean of the random variable. 14) The random variable X is the number of people who have a college degree in a randomly selected group of four adults from a particular town. Its probability distribution is given in the table. x P(X = x) 0 0.0256 1 0.1536 2 0.3456 3 0.3456 4 0.1296 A) 2.40 B) 2.00 C) 2.30 D) 2.43 Answer: A Objective: (5.2) Find Mean of Random Variable Given Probability Distribution 3 15) The random variable X is the number that shows up when a loaded die is rolled. Its probability distribution is given in the table. x P(X = x) 1 0.14 2 0.11 3 0.15 4 0.10 5 0.14 6 0.36 A) 3.94 B) 4.07 C) 3.50 D) 0.17 Answer: B Objective: (5.2) Find Mean of Random Variable Given Probability Distribution The probability distribution of a random variable is given along with its mean and standard deviation. Draw a probability histogram for the random variable; locate the mean and show one, two, and three standard deviation intervals. 16) The random variable X is the number of tails when four coins are flipped. Its probability distribution is as follows. x P(X = x) 0 1 16 1 1 4 2 3 8 3 1 4 4 1 16 µ = 2, σ = 1 A) B) 4 C) Answer: C Objective: (5.2) Draw Histogram Showing Mean and Standard Deviation Find the expected value of the random variable. 17) Suppose you buy 1 ticket for $1 out of a lottery of 1,000 tickets where the prize for the one winning ticket is to be $500. What is your expected value? A) -$0.40 B) -$1.00 C) -$0.50 D) $0.00 Answer: C Objective: (5.2) Find Expected Value of Discrete Random Variable 18) The probability distribution below describes the number of thunderstorms that a certain town may experience during the month of August. Let X represent the number of thunderstorms in August. Number of storms P(X = x) 0 0.2 1 0.2 2 0.4 3 0.2 What is the expected value of thunderstorms for the town each August? A) 2.0 B) 1.6 C) 1.5 D) 1.8 Answer: B Objective: (5.2) Find Expected Value of Discrete Random Variable 19) A contractor is considering a sale that promises a profit of $34,000 with a probability of 0.7 or a loss (due to bad weather, strikes, and such) of $16,000 with a probability of 0.3. What is the expected profit? A) $19,000 B) $18,000 C) $23,800 D) $35,000 Answer: A Objective: (5.2) Find Expected Value of Discrete Random Variable 5 Find the mean of the random variable. 20) The random variable X is the number of houses sold by a realtor in a single month at the Sendsom's Real Estate office. Its probability distribution is given in the table. x P(X = x) 0 0.24 1 0.01 2 0.12 3 0.16 4 0.01 5 0.14 6 0.11 7 0.21 A) 3.60 B) 3.40 C) 3.35 D) 3.50 Answer: A Objective: (5.2) Find Mean of Random Variable Given Probability Distribution 6