Section 5 Lab (3)

STAT 3090 S ECTIO N 5 L AB S PRI N G 2024 D ISCRETE R A N DOM V ARIABLES N AME : 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. N umber of Helicopter Transports X 0 1 2 3 4 5 6 0.1 5 0.2 0 0.3 4 0.1 9 0.0 6 0.0 5 0.0 1 or P ( X ) f X ( x ) 1

STAT 3090 S ECTIO N 5 L AB S PRI N G 2024 D ISCRETE R A N DOM V ARIABLES (a) Verify that this is a valid probability distribution. Remember you need to check two conditions. (4 points) All probabilities are between 0 < P(X=x) <1 But .15+.2+.34+.19+.06+.05+.01= 1 both conditions apply & the distribution is valid (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. 2

STAT 3090 S ECTIO N 5 L AB S PRI N G 2024 D ISCRETE R A N DOM V ARIABLES 2. WebAssign Section 5 questions 1 – 7 (29 points) ****************************************************************************** Part Two: 3. 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. (a) Joey and Chandler are playing a board game that involves rolling a pair of dice. The first person to roll doubles in this game gets ten extra points, and Joey is intent on winning. Let X = the number of rolls until he gets doubles. (2 points) No because there is no fixed number of trials (b) El Taco Loco, a local Mexican restaurant, has a challenge on their menu: If you can eat ten tacos within a 2-hour period, your meal is free. Only 3% of people typically complete the challenge. One weekend, the restaurant staff watch 100 customers attempt to eat this massive meal. Let X = the number of customers who complete the challenge. (2 points) Binomial 4

STAT 3090 S ECTIO N 5 L AB S PRI N G 2024 D ISCRETE R A N DOM V ARIABLES (c) Dwight tries to impress Angela with a card trick that involves randomly drawing 7 cards from a standard 52-card deck without replacement. (Angela is unimpressed by this.) Let X = the number of hearts she draws. (2 points) Not binomial because without replacement means it becomes dependent 4. Conveyor Belt Motors (10 points) The brushes in the motors that control the conveyor belts at Delectable Delights need to be replaced periodically. Since there are many conveyor belts through the buildings, a supply of brushes is always kept on hand. The brushes come in shipments of 1000 from a supplier who claims that 2% will be defective. Out of each shipment, a random sample of 10 brushes is 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=(1,2,3…10) n=10 p=.02 5

STAT 3090 S ECTIO N 5 L AB S PRI N G 2024 D ISCRETE R A N DOM V ARIABLES (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) P(F)10= .82 (e) Calculate the probability that a shipment will be rejected. Use probability notation and round to four decimal places. (2 points) P(X=1)= 1-P(X=0)= .18 5. Practice with a calculator (5 points) Suppose . 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. X ~ ℬ ( n = 20,   p = 0.7 ) 7

STAT 3090 S ECTIO N 5 L AB S PRI N G 2024 D ISCRETE R A N DOM V ARIABLES (a) is exactly . (1 point) P(x=15)=.179 (b) is less than 12. (1 point) P(X<12) or (X<11)=.113 (c) is greater than 12. (1 point) P(X>12) or P(X>13) or P(x<20) - P(x<12) =.772 (d) is at most 13. (1 point) P(x<13)=.392 (e) is at least 15. (1 point) P(x>15) or P(x<20) - P(x<14)= .416 6. WebAssign Section 5 questions 8 – 14 (30 points) X 15 X X X X 8