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Splash Screen. Five-Minute Check (over Lesson 11–2) CCSS Then/Now New Vocabulary Example 1: Identify and Classify Random Variables Key Concept: Probability Distribution Example 2: Construct a Theoretical Probability Distribution

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 11–2) CCSS Then/Now New Vocabulary Example 1: Identify and Classify Random Variables Key Concept: Probability Distribution Example 2: Construct a Theoretical Probability Distribution Example 3: Construct an Experimental Probability Distribution Key Concept: Expected Value of a Discrete Random Variable Example 4: Real-World Example: Expected Value Key Concept: Standard Deviation of a Probability Distribution Example 5: Real-World Example: Standard Deviation of a Distribution Lesson Menu

  3. WORK The number of hours worked in the last week by part-time employees at a fast food restaurant are 35, 20, 21, 32, 27, 12, 6, 16, 24, 19, 8, 18, 10, 28, 22, 30. Describe the center and spread of the data using either the mean and standard deviation or the five-number summary. Justify your choice. A.The data are skewed. The range of hours worked is 6 to 35. The median is 20.5 hours and the middle 50% is from 14 to 27.5 hours. B.The data are symmetric. The range of hours worked is 6 to 35. The median is 20.5 hours and the middle 50% is from 14 to 27.5 hours. C.The data are skewed. The employees work an average of 20.5 hours per week with a standard deviation of about 8 hours. D.The data are symmetric. The employees work an average of 20.5 hours per week with a standard deviation of about 8 hours. 5-Minute Check 1

  4. TICKETS The number of tickets sold for each event at a concert venue are 319, 501, 366, 648, 413, 371, 468, 399, 711, 412, 679, 702, 575, 442, 324, 578. Describe the center and spread of the data using either the mean and standard deviation or the five-number summary. Justify your choice. A.The data are symmetric. The range of ticket sales are from 319 to 711. The median is 455 and half of the ticket sales are between 385 and 613. B.The data are skewed. The range of ticket sales are from 319 to 711. The median is 455 and half of the ticket sales are between 385 and 613. C.The data are skewed. A mean of 494.25 tickets were sold with a standard deviation of about 132 tickets. D.The data are symmetric. A mean of 494.25 tickets were sold with a standard deviation of about 132 tickets. 5-Minute Check 2

  5. BOWLING John and Justin compared their bowling scores. Compare the distributions using either the means and standard deviations or the five-number summaries. Justify your choice. A.Neither distribution is skewed. The five number summary values for John’s distribution are all greater than the corresponding values of Justin’s distribution. B.Justin’s scores are skewed. The five number summary values for John’s distribution are all greater than the corresponding values of Justin’s distribution. C.John’s scores are skewed. The five number summary values for John’s distribution are all greater than the corresponding values of Justin’s distribution. D.Their scores are symmetric. The five number summary values for John’s distribution are all greater than the corresponding values of Justin’s distribution. 5-Minute Check 3

  6. RUNNING Darren and Reggie compared their times (in seconds) in the 40-meter dash. Compare the distributions using either the means and standard deviations or the five-number summaries. Justify your choice. A.Both distributions are skewed. Reggie’s median time is lower than Darren’s by two tenths of a second. The distributions have identical ranges. B.Both distributions are symmetric. Reggie’s median time is lower than Darren’s by two tenths of a second. The distributions have identical ranges. C.Both distributions are skewed. Reggie’s mean time is lower than Darren’s by almost four tenths of a second. The distributions have almost identical standard deviations. D.Both distributions are symmetric. Reggie’s mean time is lower than Darren’s by almost four tenths of a second. The distributions have almost identical standard deviations. 5-Minute Check 4

  7. SCORES Pam and Chee compared their quiz scores in Science class for the year. The middle half of the scores in Chee’s distribution are greater than the bottom 75% of scores in Pam’s distribution. The five number summary of Pam’s distribution is 68, 72, 74, 76, 100. Which of the following could be the five number summary of Chee’s distribution? A.68, 72, 81, 88, 10 B.70, 75, 80, 84, 100 C.72, 77, 79, 82, 86 D.71, 74, 82, 94, 100 5-Minute Check 5

  8. Content Standards S.MD.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). Mathematical Practices 2 Reason abstractly and quantitatively. CCSS

  9. You used statistics to describe symmetrical and skewed distributions of data. • Construct a probability distribution. • Analyze a probability distribution and its summary statistics. Then/Now

  10. random variable • experimental probability distribution • Law of Large Numbers • expected value • discrete random variable • continuous random variable • probability distribution • theoretical probability distribution Vocabulary

  11. Identify and Classify Random Variables A.Identify the random variable in the distribution, and classify it as discrete or continuous. Explain your reasoning. the number of hits for the players of a baseball team Answer:The random variable X is the number of hits, which is finite and countable, so X is discrete. Example 1

  12. Identify and Classify Random Variables B.Identify the random variable in the distribution, and classify it as discrete or continuous. Explain your reasoning. the distances traveled by the tee shots in a golf tournament Answer:The random variable X is the distance traveled, which can take on any value in a certain range, so X is continuous. Example 1

  13. Identify the random variable in the distribution, and classify it as discrete or continuous. Explain your reasoning.the number of automobiles sold by an automaker in a given week A.Continuous; the number of automobiles sold can take on any value. B.Discrete; the number of automobiles sold is finite and countable. C.Discrete; the number of automobiles sold can take on any value. D.Continuous; the number of automobiles sold is finite and countable. Example 1

  14. Concept

  15. Construct a Theoretical Probability Distribution A.X represents the sum of two cards drawn from a stack of cards numbered 1 through 8 with replacement. Construct a relative-frequency table. The theoretical probabilities associated with drawing two cards from a stack numbered 1 through 8 can be described using a relative-frequency table. When two cards are drawn, 64 total outcomes are possible. To determine the relative frequency, or theoretical probability, of each outcome, divide the frequency by 64. Example 2

  16. Construct a Theoretical Probability Distribution Answer: Example 2

  17. Construct a Theoretical Probability Distribution B.X represents the sum of two cards drawn from a stack of cards numbered 1 through 8 with replacement. Graph the theoretical probability distribution. Example 2

  18. Answer: The graph shows the probability distribution for the sum of the values on two dice X. The bars are separated on the graph because the distribution is discrete (no other values of X are possible). Each unique outcome of X is indicated on the horizontal axis, and the probability of each outcome occurring P(X) is indicated on the vertical axis. Construct a Theoretical Probability Distribution Example 2

  19. A. B. C. D. X represents the sum of two cards drawn from a stack of cards numbered 1 through 5 with replacement. Graph the theoretical probability distribution. Example 2

  20. Construct an Experimental Probability Distribution A. X represents the sum of two cards drawn from a stack of cards numbered 1 through 8 with replacement. Construct a relative frequency table for 64 trials. Draw two cards 64 times or use a random number generator to complete the simulation and create a simulation tally sheet. Calculate the experimental probability of each value by dividing its frequency by the total number of trials, 64. Example 3

  21. Construct an Experimental Probability Distribution Calculate the experimental probability of each value by dividing its frequency by the total number of trials, 64. Example 3

  22. Construct an Experimental Probability Distribution Answer: Example 3

  23. Construct an Experimental Probability Distribution B. X represents the sum of two cards drawn from a stack of cards numbered 1 through 8 with replacement. Graph the experimental probability distribution. Answer:The graph shows the discrete probability distribution for the sum of the values shown on two dice X. Example 3

  24. Concept

  25. Expected Value A tetrahedral die has four sides numbered 1, 2, 3, and 4. Find the expected value of one roll of this die. Each value has an equal chance of being rolled. E(X) = Σ[X ● P(X)] = 1(0.25) + 2(0.25) + 3(0.25) + 4(0.25) = 0.25 + 0.50 + 0.75 + 1.00 = 2.5 Answer: 2.5 Example 4

  26. RAFFLES A group passed out 2000 raffle tickets. The frequency table shows the number of winning tickets for each prize. Find the expected value of one ticket. A.$0.88 B.$1.88 C.$2.75 D.$3.75 Example 4

  27. Concept

  28. Standard Deviation of a Distribution RAFFLE At a raffle, 400 tickets are sold for $1 each. One ticket wins $100, five tickets win $10, and ten tickets win $5. Calculate the expected value and standard deviation of the distribution of winnings for a $1 ticket. Make a frequency table. Of the 400 tickets, 16 win a prize, so 400 – 16 or 384 do not. Divide the frequency by 400 to find the probability of each prize. Each ticket costs $1, so subtract 1 from each prize value. Example 5

  29. Standard Deviation of a Distribution E(X) = 0.0025(99) + 0.0125(9) + 0.0250(4) + 0.9600(–1) = 0.2475 + 0.1125 + 0.1 + (–0.96) = –0.5 Answer: The expected value of a ticket is –$0.50. Example 5

  30. RAFFLES At a raffle, 500 tickets are sold for $5 each. One ticket wins $1000, two tickets win $500, and ten tickets win $50. Calculate the expected value and standard deviation of the distribution of winnings for a $5 ticket. A.$5; 3025 B.$5; 55 C.$0; 3050 D.$0; 55 Example 5

  31. End of the Lesson

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