Question

It is desired to test if an analgesic is able to eliminate headaches one hour after its administration in at least 90% of cases. To perform this test, a sample of 30 patients was considered.

1. Construct a decision rule that completely eliminates Type I error.
2. What type of problem occurs when you have a decision rule that eliminates Type I error?
3. Build a decision rule that completely eliminates type II error.
4. What type of problem occurs when you have a decision rule that eliminates type II error?

Answer 1

a. At 90% confidence level, we need to test whether an analgesic eliminated the headaches. Here we target the population of the patients who suffered from headache. And selected a random sample of 30 patients. If we take more number of samples of same size, we will be facing a problem of rejecting the null hypothesis of no healing of headaches by the analgesic even if it is true. In case we should not have rejected the null hypothesis. We call this as a Type I error in a hypothesis testing. So we should need a decision rule to focus on the main effects that centers about the mean, not on the insignificant samples that fall far away from the mean. Let us take that the 10% of the samples will be having type I error. But our focus is 90% of the samples that we collected to test the hypothesis. The decision rule, if the critical value of the distribution of the sample at a level of significance of 0.10 alpha is less than the test statistic, and the p-value is less than the level of significance, then we can reject the null hypothesis at an risk of 10% error.

b. The problem we will be facing that they are few number of samples, and all the samples' mean value falls within the critical region. Then we will be committing the Type I error. To avoid that at least we need to take 30 such identically similar samples to make a decision about the effectiveness of the analgesic.

c. Type II error occurs when we don’t reject the null hypothesis if it is false. This is called the risk that we are taking while testing the claims or proposals. Here we need to understand the claim against the population parameter. The 1-beta or the 1- risk percentage taken is often called the power of the hypothesis testing. It decides how powerful is our claim against the null hypothesis of no effect on headaches due to the analgesic.

d. It is the risk that we are taking considering the time and money constraints in any statistical significant test would cost us or make it more expensive. At the cost of the precision to conclude the claim, we end up leaving the least contributing variables' influence on the response variable. Those least bothering events may occur in a remote chance but not frequently. That is why we feel shocked or surprised with the ground reality compared to our test proposal.