Researchers are interested in evaluating the effect of a natural product on reducing blood pressure. This will be done by comparing the mean reduction in blood pressure of a treatment (natural product) group and a placebo group using a two-sample t-test. The researchers would like to be able 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 two-sample t-test using role="math" localid="1650436089340" $\alpha =0.01$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 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 $80\%$, depending on whether the natural product is effective.

(e) would vary depending on the standard deviation of the data.

Step by step solution

01

Given information

The size of group is $50$

$\alpha =0.01$

Power$=80\%$

02

Explanation

The larger difference is easier to detect and thus has a higher power.

We'd like to see how effective a natural product is at lowering blood pressure. As a result, the smaller the difference between ${\mu }_1-{\mu }_2$

the better.

Hence ${\mu }_{1-}{\mu }_2\ge 7$if the test's power is $80\%$for alternative hypothesis.

Then, For alternate hypothesis, ${\mu }_{1-}{\mu }_2<5$the test's power will be greater than$80\%.$

Since the difference is decreased from $7$to $5$, the power will also decrease and thus the correct answer is (a).

03

Incorrect Answer 

Since the difference is decrease from 7 to 5, power would not be greater than 80%. It cannot depending on the standard deviation of the data. Therefore option b, c, d and e are incorrect answers.

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