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Q. 10

Expert-verified
Found in: Page 666

The Practice of Statistics for AP

Book edition 4th
Author(s) David Moore,Daren Starnes,Dan Yates
Pages 809 pages
ISBN 9781319113339

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.

If the researchers want to be able to detect a $5$-point difference instead, then the power of test is option (a) would be less than $80%$.

See the step by step solution

Step 1: Given information

The size of group is $50$

$\alpha =0.01$

Power $=80%$

Step 2: 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).