Suppose that you flip a coin 40 times and count the number of heads. 1.1. What is the distribution of the number of heads assuming the coin is fair? 1.2. The sample proportion of heads has an approximately normal distribution. What are the mean and standard deviation of this distribution assuming the coin is fair? 1.3. Define the Z-statistic for conducting a test of the null hypothesis that the coin is fair (i.e., has probability of a head equal to 0.5). 1.4. Suppose the experiment results in 15 heads and 25 tails. Conduct a test of the null hypothesis with type I error probability 0.05 using the normal approximation. State the Z statistic, the p-value, and the conclusion of the test (do you reject the null hypothesis or not). 1.5. If you had decided to use a type I error probability of 0.1 instead of 0.05 would your conclusion be different? Explain. 1.6. Calculate the p-value using the binomial distribution. Do you reach the same conclusion with the binomial distribution as with the normal approximation? 1.7. Calculate a 95% confidence interval for the probability of a head using the normal approximation. Does the confidence interval include the value 0.5? 1.8. Calculate a 90% confidence interval for the probability of a head using the normal approximation. How does it compare to the 95% confidence interval