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-inlabelling 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.

The following table provides the weights (in grams) of a random sample of 20 tea bags produced by machine A in the factory

5.67	5.44	5.42	5.40	5.53	5.32	5.54	5.45	5.52	5.41
5.57	5.40	5.53	5.54	5.55	5.62	5.56	5.46	5.44	5.51

(c) Another random sample of 30 tea-bags produced by machine B has the following results:

sample mean weight	sample standard deviation	Sample Proportion of underweight tea-bags
5.5063 grams	0.1208 grams	40%

Based on the sample data from both machines A and B,

(i) is there any significance evidence to show that both machines A and B have identical variances at 10% significance level? (8 marks)

(ii) construct a 95% confidence interval for the difference in the underweight proportions of machines A and B? (4 marks)

(iii) test that the company meets the legal requirement set forth on label under the assumption that both machines A and B have same variances at 1% significance level? (8 marks)

(d) What Type of error (i.e. Type I and Type II) is associated with the decision of Part (c)(iii) above? What are the possible consequences of such error for the company? (3 marks)