A pharmaceutical manufacturer is concerned about the impurity concentration in pills. It is known that from a particular production run, impurity concentrations follow a normal distribution with a standard deviation of 0.5%. A random sample of 81 pills from the production run was checked, and the sample mean impurity concentration was found to be 3.1%.
a. Assume the impurity concentration in pills should not exceed 3%. The manager asks you to test, at the 10% significance level, if there is any statistical evidence that their concern is valid.
i. State the null and alternative hypotheses.
ii. Find the critical value of the decision rule for your test.
iii. Find the p-value of your test.
iv. State your conclusion.
b. Assume the impurity concentration in pills should be exactly 3%. The manager asks you to test, at the 10% significance level, if there is any statistical evidence to suggest that their concern is valid.
i. State the null and alternative hypotheses.
ii. Find the critical value for the decision rule of your test.
iii. Find the p-value of your test.
iv. State your conclusion.
c. Compare the p-values in part (a) and part (b). Briefly explain your answer.