the number of cars passing through an intersection in a given time interval
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1 Identify the random variable in each distribution, and classify it as discrete or continuous. Explain your reasoning. the number of stations in a cable package The random variable X is the number of stations in a cable package. The stations are countable, so X is discrete. The random variable X is the number of stations in a cable package. The stations are countable, so X is discrete. the number of cars passing through an intersection in a given time interval The random variable X is the number of cars passing through an intersection. The cars are countable, so X is discrete. The random variable X is the number of cars passing through an intersection. The cars are countable, so X is discrete. Identify the random variable in each distribution, and classify it as discrete or continuous. Explain your reasoning. the number of texts received per week The random variable X is the number of texts per week. The texts are countable, so X is discrete. The random variable X is the number of texts per week. The texts are countable, so X is discrete. the height of a plant after a specific amount of time The random variable X is the height of a plant. Height can be anywhere within a certain range. Therefore, X is continuous. The random variable X is the height of a plant. Height can be anywhere within a certain range. Therefore, X is continuous. CCSS PERSEVERANCE the number of winners for 3200 hypothetical players. esolutions Manual - Powered by Cognero Page 1
2 a. Construct a relative-frequency table showing the theoretical probability. b. Graph the theoretical probability distribution. c. Construct a relative-frequency table for 50 trials. d. Graph the experimental probability distribution. e. Find the expected value. f. Find the standard deviation. a. There are 3200 total outcomes. List each possible prize in the first column. The relative frequency is the number of winners divided by This is also the associated probability P(X). List this in the second column. b. The graph shows the probability distribution for the prize amounts X. The bars are separated on the graph because the distribution is discrete (no other values of X are possible). esolutions Manual - Powered by Cognero Page 2
3 c. Use a random number generator to complete the simulation and create a simulation tally sheet. Let 1-35 represent $100, represent $250, represent $500, and so on. On your graphing calculator, select randint(1, 100, 50) for 50 trials. Then determine the frequency and relative frequency. The relative frequency is the frequency divided by the number of trials, 50. d. esolutions Manual - Powered by Cognero Page 3
4 e. Using the theoretical probability distribution, multiply each prize value by the corresponding relative frequency P (X). Then find the sum of those values. The expected value is $ f. Using the table from part e and E(X) = 922.5, subtract the expected value from each outcome. Square each difference. Then multiply by each corresponding probability. Finally, find the sum of these values. The values in the last column are rounded to the nearest hundredth. The standard deviation is about esolutions Manual - Powered by Cognero Page 4
5 esolutions Manual - Powered by Cognero Page 5
6 e. $ f RAFFLES The French Club sold 500 raffle tickets for $1 each. The first prize ticket will win $100, 2 second prize tickets will each win $10, and 5 third prize tickets each win $5. a. What is the expected value of a single ticket? b. Calculate the standard deviation of the probability distribution. c. DECISION MAKING The Glee Club is offering a raffle with a similar expected value and a standard deviation of 2.2. In which raffle should you participate? Explain your reasoning. a. List each prize value X along with the corresponding relative frequency P(X). Note that each raffle ticket costs $1, so the actual prize values are the listed values minus $1. The probability of not winning a prize is one minus the sum of the probabilities of all of the other prizes. Find X P(X). Then find the sum of those values. The expected value is $0.71. b. Using the table from part a and E(X) = 0.71, subtract the expected value from each outcome. Square each difference. Then multiply by each corresponding probability. Finally, find the sum of these values. The values in the last column are rounded to the nearest hundredth. esolutions Manual - Powered by Cognero Page 6
7 c. The standard deviation of the probability distribution for the Glee Club raffle is about half the standard deviation for the French Club raffle, so the Glee Club raffle is less risky. Since they have similar expected values, the riskier raffle will also have the potential to win more, so both raffles have good and bad qualities. It is up to the individual participant to decide which one to choose. a b c. Sample answer: The standard deviation of the probability distribution for the Glee Club raffle is about half the standard deviation for the French Club raffle, so the Glee Club raffle is less risky. Since they have similar expected values, the riskier raffle will also have the potential to win more, so both raffles have good and bad qualities. It is up to the individual participant to decide which one to choose. esolutions Manual - Powered by Cognero Page 7
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