Suppose that the drying time for a certain type of paint under specified test conditions is known to be normally distributed with mean 65 minutes and standard deviation 5 minutes. Suppose also that chemists have devised a new additive that they hope will reduce the mean drying time (without changing the standard deviation). A test is then conducted to measure the drying time for a test specimen, and the company executives decide that they will be convinced that the additive is effective only if the drying time on this specimen is less than 60 minutes.
a. If the additive actually has no effect at all on the drying time, what is the probability the company executives will mistakenly conclude that it is effective? Include a sketch with your calculation.
b. If you want to alter the cutoff value from 60 in order to reduce the error probability in part a to .05, what cutoff value should you choose?
c. Now suppose that the standard deviation of the drying times to 65 minutes is 2 rather than 5 minutes. Without doing any new calculations, describe how this change would affect your answers to parts a and b. Give an intuitive explanation for your reasoning in both cases.