A pharmaceutical company manufactures large batches of analgesic caplets which are designated to contain 200 mg of aspirin. The concentration of aspirin per caplet is actually a random variable X normally distributed about a mean 200 mg with standard deviation known to be 7.5 mg when the manufacturing process is in "control". As a Quality Control exercise small random samples of n caplets are taken from each production run and tested for potency.
A random sample of n = 16 caplets is selected with a view to testing the hypotheses:
H0: m = 200 mg H1: m ≠ 200 mg
Assuming a type I error of 5%, determine the critical region(s) for this test in terms of the sample mean.
What is the probability of a type II error for this test if the true mean is in fact 195 mg?
What impact would reducing the sample size have on the type I and type II errors?
The observed mean for the sample of 16 observations was found to be 203 mg. Calculate the "p-value" associated with this sample statistic. Based on the stated hypotheses, what do you conclude? Would your conclusion change if a = 10%?
Assuming a significance level of 5% is to be used, what size sample should be taken if it is required that we reject H0 if the true mean deviates from 200 mg by 5 mg or more with power of at least 90%?