Chapter 10: Q. T10.10 (page 699)

Researchers want to evaluate the effect of a natural product on reducing blood pressure. They plan to carry out a randomized experiment to compare the mean reduction in blood pressure of a treatment (natural product) group and a placebo group. Then they will use the data to perform a test of $H_{0} : μ_{T} - μ_{P} = 0$ versus $H_{a} : μ_{T} - μ_{P} > 0$ , where $μ_{T}$ = the true mean reduction in blood pressure when taking the natural product and μP = the true mean reduction in blood pressure when taking a placebo for subjects like the ones in the experiment. The researchers would like to detect whether the natural product reduces blood pressure by at least $7$ points more, on average, than the placebo. If groups of size $50$ are used in the experiment, a twosample t test using $α = 0.01$ will have a power of $80 %$ to detect a $7$ -point difference in mean blood pressure reduction. If the researchers want to be able to detect a $5$ -point difference instead, then the power of the test
a. would be less than $80 %$ .
b. would be greater than $80 %$ .
c. would still be $80 %$ .
d. could be either less than or greater than $80 %$ .
e. would vary depending on the standard deviation of the data

Short Answer

Expert verified

Option(a) is correct answer. The power of test would be less than $80 %$

Step by step solution

Given information

We need to find the test power.

Simplify

According to the question, if a researcher wishes to discover a seven-point difference, he or she will require a power of $80$ percent.
Then, if a study wants to discover a five-point difference, we know that larger differences are easier to detect and hence have a higher power.
As a result, if the difference is reduced from $7$ to $5$ , the power is reduced as well, and the right answer is (a), implying that the test's power is less than $80 %$ .