A hockey team is convinced that the coin used to determine the order of play is weighted for tails. The team captain steals this special coin and flips it 19 times to evaluate the hypothesis that the coin is weighted for tails, and it shows up tails 11 times. Test this hypothesis (use alpha=.01). 1. What is the appropirate test? I think this is a z test am I correct? 2. Are these the correct null and alternative hypotheses? H0: p = 0.5 Ha: P > 0.5 3. Find the critical value(s). I don't know how to find this can you please show me. 4. Calculate the obtained statistics. I do not know how to find this can you please show me how? 5. Make a decision: (can you explain to me which one of these it would be and why?) Reject the null. Retain the alternative. Retain the null. Reject the alternative 6. What does your decision mean? Please explain what it means. A hockey team is convinced that the coin used to determine the order of play is weighted. The team captain steals this special coin and flips it 19 times to evaluate the hypothesis that the coin is weighted, and it shows up heads 12 times. Test this hypothesis (use alpha=.05).
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Just Keep Flipping Have you ever wondered how fair a coin is? Perhaps if you flipped a coin enough times, you'd get 20 heads in a row. Or 30? Probably not. But in Chapter 20, we discuss errors in testing. And we learned that Type I errors, with a significance level of 5%, will generally happen ... 5% of the time. That means that if you're willing to keep trying, you're bound to mess up eventually. Let's try to do just that! 1. First, examine when we would reject fairness for flipping a coin 100 times. We will be simulating 100 coin flips and will run a 2-sided test to determine if the coin is unfair. • What are the null and alternate hypotheses? • What is the standard deviation for this test? • What is the critical value for this test? Or, said differently, what z-score corresponds to a 5% significance level, at which point we reject H₀? • Using the standard deviation and the critical value, calculate how many (or how few) times you would have to flip heads to reject the H₀. 2. Now simulate 100 coin flips by using the Random Binomial generator on your favorite technology tool. Use n = 1, p = 0.5 and x = 100. You should generate a list of 100 0's and 1's. Sum the list and you will find the simulated percentage of "heads". Repeat this process 40 times and graph the results. 3. Examining your graph, did you find an extreme result? Extreme enough to reject the fairness of your coin? Given that we were using a 5% significance level, how many rejections did you expect to get? What does tell you about the problem with running significance tests over and over again?
Dominador T.
Problem 5: Another Version of Skunk Redux uses a single-sided die with two faces labeled O and two faces labeled 1. So that each comes up with a probability of 1/3. If you are standing when the die is rolled, then if a 2 comes up, you add that to your current score. But if a 1 comes up, the game is over for you and your score is 0. If you leave the game before it is thrown, your payoff is your score at that point. Calculate the optimal strategy: Calculate the average payoff obtained using the strategy in (a). The game is simple enough that you can simply make a list of all possible scenarios and take the weighted average of their payoffs. It's useful to group together (without listing) all scenarios which give you zero payoff. Calculate the "tug" payoff for the strategy "exit after k rolls if the game is not already over for k values k >= 3" and compare with the answer to (b). You might do this by making a list of all possible scenarios, but for k >= 3 it's worth trying to find a more efficient method. Particularly anticipating (0) below. Calculate the Average Payoff for the strategy "exit after k rolls if the game is not already over" for the general value k.
Sri K.
Imagine that you flip a fair coin 20 times. This will create a sequence of heads and tails, and there are lots of possible sequences (just over one million equally likely microstates). Now, imagine looking for strings of consecutive H's or consecutive T's within each sequence of 20 flips. For instance, one sequence (one microstate) is THHTTTTHTHHHTTHHTTHT, which contains one string of four consecutive T's, one string of 3 consecutive H's, and many strings of two consecutive H's and T's. Let's focus on the LONGEST string of consecutive H's or T's in each 20-coin sequence and define the LENGTH of the longest string to be the "macrostate" for that particular microstate. In the example above, the macrostate is "4", because the longest string is a string of four consecutive T's. (a) How many microstates are there for macrostate = 1? Explain. How many microstates are there for macrostate = 20? Explain. How many microstates are there for macrostate = 19? Explain. (b) What are the five most likely macrostates? What is the probability that each of these macrostates occurs? Part (a) can be done without any computational tools, and it helps to list the possible microstates as part of your explanation. However, part (b) is different. We cannot list every microstate (every possible sequence), because we cannot create a well functioning worksheet with over one million rows, so instead, we will simulate the results with a much smaller number of trials in order to estimate the answers to the questions in part (b). Submit your Excel workbook to Moodle to be graded, i.e., do NOT submit a PDF with values and formulas.
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