Solved: The concentration of an active ingredient in the

Chapter 9, Problem 8.70

(choose chapter or problem)

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 \(\sigma = 5%\). A sample of outputs from 30 independent experiments gives the average concentration of \(\overline{x}_A = 64.5%\).

(a) Does this sample information with an average concentration of \(\overline{{x}_A = 64.5%\) provide disturbing information that perhaps \(\mu_A\) 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 \(\overline{x}_B\) turns out to be 70%. Comment on whether or not the sample information on catalyst B strongly suggests that \(\mu_B\) is truly greater than \(\mu_A\). Support your answer by computing

\(P\left(\bar{X}_{B}-\bar{X}_{A}\ \geq\ 5.5 \mid\ \mu_{B}=\mu_{A}\right)\)

(c) Under the condition that \(\mu_A = \mu_B = 65%\), give the approximate distribution of the following quantities (with mean and variance of each). Make use of the Central Limit Theorem.

i) \(\overline{X}_B\);

ii) \(\overline{X}_A - \overline{X}_B\);

iii) \(\frac{\overline{X}_A - \overline{X}_B}{\sigma \sqrt{2/30}}\).