Critical value and hypothesis testing

(a) What is the critical value and how is the critical value used in hypothesis testing?

(b) Look at the example in the attachment about the new drug, Releeva. In the Releeva example, the sample size is sufficiently large (100 patients), thus we can assume normality and use a Z statistic for the hypothesis testing procedure. How does this change if the sample size is 25 patients?

Let's say that a drug company wants to show that its new drug, Releeva, provides pain relief faster than the 30 minutes its well-established competitor, No-ache, advertises. The Releeva team issues the drug to 100 people in a pain relief clinic and records the time to relief reported by the patients.

Step 1

The null hypothesis, or status quo, is that the mean time to relief is 30 minutes (or more). If the data causes us to reject this hypothesis, then the alternate (which supports our position) must be true. The alternate hypothesis is that the mean time to relief is less than 30 minutes. In symbols, it would look like this:
Ho: μ; ≥ 30
Ha: μ; < 30

Note that the equal sign is in the null hypothesis. Also, this is a "one-tail test" because the alternate hypothesis only tests one side of the normal curve. Had the question asked if Releeva had a mean time to relief of exactly 30 minutes, the test would have been:
Ho: μ; = 30.
Ha: u < > 30

In this "two-tailed test" the alternate hypothesis looks to both tails of the distribution because rejecting the null hypothesis means the time to relief is either less than 30 minutes or more than 30 minutes.

Step 2

We decide that we want to be 95% confident in our answer, α is .05.

Step 3

In our Releeva example, because the sample size is sufficiently large (100 patients), we can assume normality and use a Z statistic.

Step 4

The decision rule would be to reject the null hypothesis if the calculated value of Z is greater than the critical value. By consulting a Z table, the Z value for a one-tail test associated with α of .05 is 1.645. Since we are testing "less than", we are looking at the left side or negative side of a normal distribution. Therefore, the critical value of Z is -1.645.

Step 5

Our job now is to calculate the value of Z based on a sample of patients who used Releeva. If the calculated value of Z is larger than (more negative than) -1.645, then the result will fall in the rejection region, and we will reject the null hypothesis in favor of the alternate. This would indicate that Releeva is faster and that the observed results are probably not due to the random chance that the individuals selected for sampling were predisposed to a favorable result. We would say there is statistically significant evidence that the new drug works faster. On the other hand, if the calculated value of Z is smaller than -1.645 (less negative), that would indicate that observed result was probably due to chance rather than the efficacy of Releeva. The result in that case would be deemed not statistically significant.