5 assessing garys models games with gary our friend

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5. Assessing Gary's Models Games with Gary Our friend Gary comes over and asks us to play a game with him. The game works like this: We will flip a fair coin 10 times, and if the number of heads is greater than or equal to 5, we win! Otherwise, Gary wins. We play the game once and we lose, observing 1 head. We are angry and accuse Gary of cheating! Gary is adamant, however, that the coin is fair. Gary's model claims that there is an equal chance of getting heads or tails, but we do not believe him. We believe that the coin is clearly rigged, with heads being less likely than tails. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Running tests --------------------------------------------------------------------- Test summary Passed: 1 Failed: 0 [ooooooooook] 100.0% passed Out[37]: 8.2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Running tests --------------------------------------------------------------------- Test summary Passed: 1 Failed: 0 [ooooooooook] 100.0% passed
Question 1 Assign coin_model_probabilities to a two-item array containing the chance of heads as the first element and the chance of tails as the second element under Gary's model. Make sure your values are between 0 and 1. In [39]: coin_model_probabilities = make_array( .5 , .5 ) coin_model_probabilities In [40]: _ = ok . grade( 'q5_1' ) Question 2 We believe Gary's model is incorrect. In particular, we believe there to be a smaller chance of heads. Which of the following statistics can we use during our simulation to test between the model and our alternative? Assign statistic_choice to the correct answer. 1. The distance (absolute value) between the actual number of heads in 10 flips and the expected number of heads in 10 flips (5) 2. The expected number of heads in 10 flips 3. The actual number of heads we get in 10 flips In [41]: statistic_choice = 1 statistic_choice Question 3 Define the function coin_simulation_and_statistic , which, given a sample size and an array of model proportions (like the one you created in Q1), returns the number of heads in one simulation of flipping the coin under the model specified in model_proportions . Hint: Think about how you can use the function sample_proportions . In [42]: def coin_simulation_and_statistic (sample_size, model_proportions): return sample_proportions(sample_size,model_proportions) . item( 0 ) * sample_size coin_simulation_and_statistic( 10 , coin_model_probabilities) In [43]: Out[39]: array([0.5, 0.5]) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Running tests --------------------------------------------------------------------- Test summary Passed: 1 Failed: 0 [ooooooooook] 100.0% passed Out[41]: 1 Out[42]: 4.0
_ = ok . grade( 'q5_3' ) Question 4 Use your function from above to simulate the flipping of 10 coins 5000 times under the proportions that you specified in problem 1. Keep track of all of your statistics in coin_statistics . ) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Running tests --------------------------------------------------------------------- Test summary Passed: 1 Failed: 0 [ooooooooook] 100.0% passed
Let's take a look at the distribution of statistics, using a histogram. Running tests --------------------------------------------------------------------- Test summary Passed: 1 Failed: 0 [ooooooooook] 100.0% passed In [64]: #Draw a distribution of statistics Table() . with_column( 'Coin Statistics' , coin_statistics) . hist()
Question 5 Given your observed value, do you believe that Gary's model is reasonable, or is our alternative more likely? Explain your answer using the distribution drawn in the previous problem.

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