Question I To settle a debate over whether Coca Cola or Pepsi Cola is more popular, a random sample of 79 participants was selected and asked to state theil preference for one of the drinks. 43 of them preferred Coca Cola. Let p denote the proportion of the population who prefer Coca Cola. (a) (b) (c) (d) Compute the probability tllat a participant selected at random from the sample Will prefer Coca Cola. (I mark) What is the probability that nvo panicupants selected (without replacement) at random from the sample will both prefer Coca Cola? (3 marks) Constmct a confidence interval for p. (4 marks) Constmct a one-sided confidence interval for p of the fonn (r, I) (3 marks) What conclusion about p can be drawn from these confidence mtenals? In particular, can we claim that Coca Cola is the more popular drink? That IS, is p > 0.50? (3 marks) Suppose the sample size is increased to 7,900 participants (100 times), and the sample proportion stays the same at (g) What impact does the bigger sample have, if any, on the standard deviation of P? (3 marks) Will the enlarged sample size affect your answer in Question I(e)? If So, m what (3 marks) Question 2 The two coms A and B look identical Coin A a fail coin For coin B, the probability of obtaining a head is and the probability of obtaimng a tail 60% in a toss. Your friend Amina picked one of the coins at random and wanted to find out whether it was coin A or com B. She tossed the selected coin that 39 times and obtain 18 heads. She asked you to help her analyse the data. You decide to use hypothesis testing using 5% as the level of significance. Your null hypothesis is that coin A was picked. That is, the probability p of getting a head is Null hypothesis, Ho: coin A was picked, p = 0.50. (a) (b) (c) (d) (e) Wnte down the altemahve hypothesi s. (3 marks) Let X denote the number of heads obtained m 39 tosses. Under the null hypothesis what IS the dish-ibution of X? (3 marks) Under the null hypothesis, what the expected value and variance of the sample proportion ? That is, compute E (P) and var(). (4 marks) What test statistic myuld you use for the test? (3 marks) Is the null hypothesis rejected at the 5% level of significance ? (4 marks) Which IS typically considered to be the more serious enor between Type I and Type II (3 marks)