Section 5 Lab

STAT 3090 S ECTION 5 L AB S PRING 2024 D ISCRETE R ANDOM V ARIABLES N AME : Cam Allen P URPOSE : This lab looks at how to calculate probabilities, means and standard deivations for discrete random variables. It will reinforce the probability rules you have learned and introduce you to the special discrete random variable known as the Binomial Random Variable. O BJECTIVES : Upon successful completion of this activity, you will be able to…  Classify a random variable as discrete or continuous,  Compute probabilities and answer relevant questions from a table of values.  Understand the characteristics of a Binomial Random Variable  Calculate probabilities and the mean and standard deviation of a Binomial Random Variable Part One: 1. Helicopter Transports (20 points) The 4077 th Mobile Army Surgical Hospital is considering the purchase of a helicopter to transport critical patients. The relative frequency of X , the number of times a helicopter is used to transport critical patients each month, is determined from a similarly-sized army hospital as given by the following probability distribution. Number of Helicopter Transports X 0 1 2 3 4 5 6 f X ( x ) or P ( X ) 0.15 0.20 0.34 0.19 0.06 0.05 0.01 (a) Verify that this is a valid probability distribution. Remember you need to check two conditions. (4 points) This is a valid probability distribution because P(X) is less than or equal 1 and is greater than or equal to 0, while the summation of P(X) = 1. (b) Find the mean number of times the helicopter is used to transport critical patients each month. (4 points) To get full credit, you need to: - Use appropriate symbol for the mean - Show your work by plugging values into the formula. - Get a correct answer. - Include units. μ = (1*0.2) + (2*0.34) + (3*0.19) + (4*0.06) + (5*0.05) + (6*0.01) = 2 times 1

STAT 3090 S ECTION 5 L AB S PRING 2024 D ISCRETE R ANDOM V ARIABLES (c) Find the standard deviation of the number of times the helicopter is used to transport critical patients. (4 points) To get full credit, you need to: - Use appropriate symbol for the standard deviation. - Show your work by plugging values into the formula. - Get a correct answer and round it to two decimal places. - Include units. σ = sqrt ((1*0.2) + (4*0.34) + (9*0.19) + (16*0.06) + (25*0.05) + (36*0.01) - 4) = 1.36 times (d) Find the probability that a helicopter will not be used at all to transport patients. Use probability notation. (4 points) P(0) = 0.15 (e) Find the probability that a helicopter will be used at least once to transport critical patients. Use probability notation. (4 points) 1 – P(X < 0) = 1 – 0.15 = 0.85 Part Two: 2. Identify a Binomial Random Variable (6 points) Are the following random variables binomial random variables? - If yes, state the possible values of the random variable, and give the parameters n and p . - If not, state why, i.e., which of the four characteristics of a binomial experiment given in lecture guide is violated. 2

STAT 3090 S ECTION 5 L AB S PRING 2024 D ISCRETE R ANDOM V ARIABLES selected for testing. If there are any defective brushes in the sample, the shipment will be rejected. Note that all 10 brushes are tested. Let X be the number of defective brushes in a sample of size 10. (a) X can be treated as a binomial random variable. Write out the possible values of X . Determine its parameters n and p . (2 points) X = 0, 1, 2, … , 10. N = 10, p = 0.02. (b) Calculate the mean for X . (2 points) To get full credit, you need to: - Use appropriate symbol for mean. - Show your work by plugging values into the formula. - Get a correct answer. - Include units. μ = 10 * 0.02 = 0.2 defectives (c) Calculate standard deviation for X . (2 points) To get full credit, you need to: - Use appropriate symbol for standard deviation. - Show your work by plugging values into the formula. - Get a correct answer and round it to two decimal places if necessary. - Include units. σ = sqrt (10 * 0.02 * (1 – 0.02)) = 0.44 defectives (d) Calculate the probability that a shipment will NOT be rejected. (2 points) To get full credit, you need to: - Use probability notations - Show your work by plugging values into the formula. DO NOT show calculator jargon. - Get a correct answer and round it to four decimal places. P(X=0) = 10C0 (0.02) ^0 * (1 – 0.02)^10 = 0.82 4

STAT 3090 S ECTION 5 L AB S PRING 2024 D ISCRETE R ANDOM V ARIABLES (e) Calculate the probability that a shipment will be rejected. Use probability notation and round to four decimal places. (2 points) P(X > 0) = 1 – P(X=0) = 1 – 0.82 = 0.18 4. Practice with a calculator (5 points) Suppose X B ( n = 20 , p = 0.7 ) . Use a calculator to find the probabilities of the following events occurring. To get full credit, you need to: - Use probability notation. - Get correct answers and round them to four decimal places. - No need to show work. (a) X is exactly 15 . (1 point) P(X=15) = 0.1789 (b) X is less than 12. (1 point) P(X<12) = 0.1137 (c) X is greater than 12. (1 point) P(X>12) = 0.7727 (d) X is at most 13. (1 point) P(X<=13) = 0.3920 (e) X is at least 15. (1 point) P(X>=15) = 0.4164 5. WebAssign Section 5 (59 points) 5