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 (XA bar ) = 64.5%.
(a) Does this sample information with an average concentration of (XA bar) = 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 (XB bar) 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( (XB bar) - (XA bar) ) ≥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) (XB bar) ;
ii) (XA bar) - (XB bar);