iii) Let B be a random variable with the Binomial(n, p) distribution. Using the central limit theorem and the fact that B has the same distribution as X1 + . .. + Xn. where X1, ..., Xn are independent and each has the Bernoulli(p) distribution, show that B - np Vup(1 - p) has approximately a standard normal distribution for large n. [6 marks] iv) Someone claims to be able to predict the outcome of a coin flip. In n = 100 flips they are correct 55 times. Let B be a random variable counting the number of successful predictions in 100 flips and p be the probability they predict a single flip correctly. Test the hypothesis Ho : p = 1/2 against the alternative H1 : p > 1/2 at significance level o = 0.05 using the test statistic B - 100p T = V100p(1 - P) which you can assume to have approximately the N(0, 1) distribution. [6 marks] v) What is the minimum number of successful predictions out of 100 flips which would cause you to reject the null hypothesis in the test of part iv)? [4 marks]