I have work for all of them but I just want to reconfirm my thoughts with a tutor.

Suppose you are a casino owner, and your casino runs a very simple game of chance. The dealer flips a coin. The customer wins$9 from the casino if it comes up heads and loses$10 if it comes up tails.

Question 1.Assuming no one is cheating and the coin is fair, if a customer plays twice, what is the chance they make money?

A certain customer plays the game 20 times and wins 13 of the bets. You suspect that the customer is cheating! That is, you think that their chance of winning is higher than the normal chance of winning.

You decide to test your hunch using the outcomes of the 20 games you observed.

Question 2.Define the null hypothesis and alternative hypothesis for this investigation.

Question 3.Given the outcome of 20 games, which of the following test statistics would be a reasonable choice for this hypothesis test?

Hint: For a refresher on choosing test statistics, check outSection 11.3 of the textbook.

1. Whether there is at least one win.
2. Whether there is at least one loss.
3. The number of wins.
4. The number of wins minus the number of losses.
5. The total variation distance between the probability distribution of a fair coin and the observed distribution of heads and tails.
6. The total amount of money that the customer won.

Assignreasonable_test_statisticsto alistof numbers corresponding to these test statistics.

Question 4. Simulate a function calledsimulatethat simulates your test statistic. It should take no arguments. It should return the number of wins in 20 games simulated under the assumption that the result of each game is sampled from a fair coin (one that is equally likely to get heads or tails).

Question 5.Using 10,000 trials, generate simulated values of the number of wins in 20 games. Assigntest_statistics_under_nullto an array that stores the result of each of these trials.

Question 6.Using the results from Question 5, generate a histogram of the empirical distribution of the number of wins in 20 games.

Question 7.Compute an empirical P-value for this test.

Hint:Which values of our test statistic are in the direction of the alternative hypothesis?

Question 8.Suppose you use a P-value cutoff of 1%, according to the arbitrary conventions of hypothesis testing. What do you conclude about the null and alternative hypotheses? Why?

Question 9.Isp_valuethe probability that the customer cheated, or the probability that the customer didn't cheat, or neither? If neither, what is it?

Question 10.Is 1% (the P-value cutoff) the probability that the customer cheated, or the probability that the customer didn't cheat, or neither? If neither, what is it?

Question 11.Suppose you run this test for 400 different customers after observing each customer play 20 games. When you reject the null hypothesis for a customer, you accuse that customer of cheating. If no customer were actually cheating, can we compute how many we will incorrectly accuse of cheating? If so, what is the number? Explain your answer. Assume a 1% P-value cutoff.