The concentration of an active ingredient in the output of a chemical reaction is strongly influenced by the catalyst that is used in the reaction. It is felt that when catalyst A is used, the population mean concentration exceeds 65%. The standard deviation is known to be σ=5%. A sample of outputs from 30 independent experiments gives the average concentration of x¯¯¯A=64.5%.
(a) Does this sample information with an average concentration of x¯¯¯A=64.5% provide disturbing information that perhaps μA is not 65%, but less than 65 %? Support your answer with a probability statement.
(b) Suppose a similar experiment is done with the use of another catalyst, catalyst B. The standard deviation σ is still assumed to be 5% and x¯¯¯B turns out to be 70%. Comment on whether or not the sample information on catalyst B strongly suggests that μB is truly greater than μA. Support your answer by computing P(X^B−X^A≥5.5|μB=μA).
(c) Under the condition that μA=μB=65%, give the approximate distribution of the following quantities (with mean and variance of each). Make use of the Central Limit Theorem.
i) X^B;
ii) X^A−X^B;
iii) X^A−X^Bσ2/30−−−−√.