Question 2 Suppose you have a fair coin (a coin is considered fair if there is an equal probability of being heads or tails after a flip). In other words, each coin flip i follows an independent Bernoulli distribution X Ber(1/2). Define the random variable X, as: i if coin flip i results in heads 10 if coin flip i results in tails a. Suppose you flip the coin n = 10 times. Define the number of heads you observe as the random variable Y10 = x and note that Yio follows a Binomial distribution: Y1o Bin(10,1/2) __ » __ __ __ __ _ __ _ with f(t) = P(Yn = y) = (*)p”(1 p)(n-y). Calculate the cumulative probability that you will observe a total 0, 1, or 2 heads in the 10 coin flips. b. Now suppose that you don’t know if the coin is fair or not, but you would like to find out. To do this, you will form a hypothesis about the true nature of the coin (the probability that it will be heads) and look to see if there is enough evidence to reject that hypothesis. Specifically, we want to test the hypothesis that p= 1/2 (the coin is fair). We will call this the null hypothesis. Next we need to consider what kind of evidence will lead us to reject this null hypothesis. We do this by defining an alternative hypothesis, which is what we would expect to be true if the null hypothesis is false. In this case, we will define the alternative hypothesis ns p < 1/2. In other words, we will decide that the null hypothesis is false if we determine that there is enough evidence that the alternative is correct. We can express our hypothesis as: Ho :p = 1/2 H:P < 1/2 where Ho and H signify, our null and alternative hypotheses, respectively. You test the hypothesis by performing an experiment: Flip the coin ten times and count the number of heads that occur. Suppose that the outcome of this experiment is that you observe two heads out of ten flips (Y10 = 2). Given your answer to part (a) above, which hypothesis do you believe is true, Ho or Hi? Explain how you reached your conclusion,
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