A production line produces goods, and each good is defective indepen- dently with a probability of . (a) Suppose that the production line is shut down immediately after 60 defective goods have been produced. Find the expectation for the total number of goods produced, proving any formulae used. (b) Quoting a relevant formula, find the standard deviation for the number of non-defective goods produced. (c) State the Central Limit Theorem and explain why it might be useful for investigating cumulative probabilities for the type of distribution that you used in a) and b). (d) Applying Central Limit Theorem (with justification), calculate the approximate probability that between 1100 and 1200 non- defective goods (inclusive) are produced. (e) Suppose that instead of shutting down after 60 defective goods, the factory just produces exactly 1000 goods. Applying Central Limit Theorem (with justification), calculate the approximate probabil- ity that between 45 and 55 defective goods (inclusive) are pro- duced. Then calculate the approximate probability that more than 70 defective goods are produced. Hint: Refer to a normal distribution table or calculator.