The chemical benzene is highly toxic to humans. However, it is used in the manufacture of many medicine dyes, leather, and many coverings. In any production process involving benzene, the water in the output of the process must not exceed 7950 parts per million (ppm) of benzene because of government regulations. For a particular process of concern the water sample was collected by a manufacturer 25 times randomly and the sample average x was 7960 ppm. It is known from historical data that the standard deviation a is 100 ppm.

(a) What is the probability that the sample average in this experiment would exceed the government limit if the population mean is equal to the limit? Use the central limit theorem.

(b) Is an observed x = 7960 in this experiment firm evidence that the population mean for the process exceeds the government limit? Answer your question by computing P(X > 7960 | p = 7950). Assume that the distribution of benzene concentration is normal.