Question

It is well known that a placebo, a fake medication or treatment, can sometimes have a positive effect just because patients often expect the medication or treatment to be helpful. An article gave examples of a less familiar phenomenon, the tendency for patients informed of possible side effects to actually experience those side effects. The article cited a study in which a group of patients diagnosed with benign prostatic hyperplasia is randomly divided into two subgroups. One subgroup of size 55 received a compound of proven efficacy along with counseling that a potential side effect of the treatment is erectile dysfunction. The other subgroup of size 52 is given the same treatment without counseling. The percentage of the no-counseling subgroup that reported one or more sexual side effects is 19.23%, whereas 41.82% of the counseling subgroup reported at least one sexual side effect. State and test the appropriate hypotheses at significance level 0.05 to decide whether the nocebo effect is operating here. [Note: The estimated expected number of "successes" in the no-counseling sample is a bit shy of 10, but not by enough to be of great concern (some sources use a less conservative cutoff of 5 rather than 10).]

State the relevant hypotheses. (Use p₁ for the true proportion of patients experiencing one or more sexual side effects when given no counseling and p₂ for the true proportion of patients experiencing one or more sexual side effects when receiving counseling that a potential side effect of the treatment is erectile dysfunction.)

Calculate the test statistic and P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.)

z= -2.54

p-value=?

Answer 1

For sample 1, we have that the sample size is and the sample proportion

For sample 2, we have that the sample size is and the sample proportion

The value of the pooled proportion is computed as

Also, the given significance level is α=0.05.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p1​=p2​

Ha: p1​<p2​

This corresponds to a left-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a left-tailed test is = -1.64

The rejection region for this left-tailed test is

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that , it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.0057, and since , it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population proportion p1​ is less than p2​, at the 0.05 significance level.

Graphically

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