3. Application of CLT. The Central Limit theorem is an important tool in statistical analysis for simplifying questions about sums of random variables to questions about Gaussian RV. Consider a transistor manufacturing plant. To keep costs down, the manufacturing procedure can break a transistor with a 50% chance. (a) Assume that you get a batch of 100 transistors from the plant. Using the CLT. estimate the probability that the number of broken transistors is between 35 and 65. Similarly, estimate the probability that the number is between 50 and 55. (b) Repeat part (a) but with a batch of 1000 transistors and determine the probabil- ity for the the number of broken transistors being in the intervals (350.650] and (500.550], respectively. (c) Suppose that the plant decides to improve their procedures so that only 20% of transistors are broken. To confirm the improvement in the process, a large number n of transistors are sampled and a relative frequency estimate , (n) for the probability of being broken is obtained. Use the central limit theorem to estimate how many transistors n should be sampled in order that the probability is at least .95 that fa(n) differs from 0.20 by less than 0.02