stat act 7

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BA 350: Statistics 1 Activity 7 (Chapter 5) 10/2/2020 Instructions: Complete the following questions. You may discuss with classmates, but you must turn in your own assignment. Unless otherwise specified, responses should be brief (one or a few sentences). Completed assignments should be submitted to Activity 7 on Schoology. Discrete Probability Distributions 1. As of 10/01/2020, the Powerball jackpot is $43 million and it costs a mere $2 to play. The probability of winning the Powerball grand prize is 1 out of 292,201,338 (Note that the website reports this as an “odds,” but this is a bit of a misnomer because an odds is reported as the ratio of wins to losses, but this is the number of wins out of total attempts, which is really the probability of winning). Use this information to answer the following questions. a. Let . Complete the table representing the probability distribution for . Be sure to factor in the $2 you pay to play. Outcome x P(X=x) Win 42,999,998 1/292,201,338 Lose -2 0.99999999 b. What is the expected payoff for one play of the Powerball? -1.85284 c. Write a sentence interpreting the expected payoff that you found in part (b). That is, explain what this value means. This expected payoff value is the “long run” weighted average value. Therefore, the chances of winning are extremely low since it is such a small value. d. How much would the Powerball have to pay out to make you break even in terms of expected payoff? .0000000584403

A Powerball winner is determined by randomly drawing six balls with numbers on them. When you play, you choose your own six numbers, and if all six match the numbers drawn from the six balls, you win. Since the balls are chosen at random, each Powerball play is independent. Use this as well as the information already provided above to answer the following questions. e. Suppose you decide to play the Powerball once each day for a year, for a total of 365 plays. What probability distribution is appropriate to use for counting the number of times you win out of those 365 Powerball plays? Binomial distribution. f. Using the probability distribution you chose in part (e), calculate the probability of winning exactly once in the year. Show your calculation as well as the final answer. = = 0.000001 𝑛! 𝑥!(𝑛−𝑥)! π 𝑥 (1 − π) 𝑛−𝑥 1! 1!(1−0)! π 0 (1 − π) 1−0 g. Based on the same probability distribution, what is the expected number of times you will win the Powerball? Less than one time. h. Calculate the standard deviation of the number of times you will win the Powerball. Standard deviation is zero because it is such a low probability of winning even one time.

2. Open this app for exploring Poisson probability distributions. Remember that a Poisson probability distribution has only one parameter: (which equals the expected value and the variance of the distribution). Use the slider in the app to change . Describe how changes the shape of a Poisson distribution. The higher the number, the more flattened out the shape of the distribution becomes. Though it is still a bell curve, the poisson distribution begins to level out. 3. Over the summer, my wife and I bought a birdfeeder and began casually keeping track of the birds we saw each day. As a rough estimate, we saw on average 7 unique bird species at the feeder per day (we didn’t watch all day, so I’m sure we missed some!). Suppose we use a Poisson probability model to approximate the number of unique bird species we saw per day. a. Use the Poisson pmf to find the probability of us seeing exactly 7 unique bird species in a day. Show your work for this calculation . P(X=x|λ) = = = .00638 𝑒 −λ λ 𝑥 𝑥! 𝑒 −7 7 1 1! b. Bonus (optional): Think through the assumptions of a Poisson distribution and explain why a Poisson probability model would or would not be an appropriate model to use for this random variable. There’s only one variable and if you are attempting to find the average number of birds, a Poisson distribution is an appropriate choice.

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stat act 7

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