Question 1: (50 marks) A giant tea company specializing top grade products is studying one of its tea-bag filling process of its core brand which label weight on the package indicates that, on average, there are 5.5 grams of tea in a bag. A quality characteristic of interest of the investigation is the weight of the tea in the individual bags. If the bags are under-filled, two problems arise. In the first place, customers may not be able to brew the tea to be as strong as they wish. Secondly, the company may be in violation of the truth-in- labelling laws. On the other hand, if the average amount of teas in a bag exceeds the label weight, the company is giving away expensive products that will erode its profit margin. Getting an exact amount of tea in a bag is troublesome because of variation in the temperature and humidity inside the factory, differences in the density of the tea, and the extremely fast filling operation of its two machines A and B which contribute 40% and 60%, respectively, of the total outputs. (a) The process manager would like to take a random sample to study the case. (1) If the manager would like to estimate the proportion of its tea-bags that is underweight (i.e. less than label weight of 5.5 grams) with a maximum error of 1% at a 90% confidence level, what sample size is required? (3 marks) (ii) Believing that the standard deviation of the weight of tea-bags is 0.1 grams, the manager would like to estimate the average weight of its tea-bags with a maximum error of 0.01 grams at a 95% confidence level. How large a sample should be taken in this case? (3 marks) (iii) If the manager would like to take ONE sample to achieve both of the objectives (i) and (ii), what sample size would you recommend? Briefly explain your recommendation