Question

Question: Testing the placebo effect. The placebo effect describes the phenomenon of improvement in the condition of a patient taking a placebo—a pill that looks and tastes real but contains no medically active chemicals. Physicians at a clinic in La Jolla, California, gave what they thought were drugs to 7,000 asthma, ulcer, and herpes patients. Although the doctors later learned that the drugs were really placebos, 70% of the patients reported an improved condition. Use this information to test (at α = 0.05) the placebo effect at the clinic. Assume that if the placebo is ineffective, the probability of a patient’s condition improving is 0.5.

Short answer

It is concluded that the placebo is ineffective; the probability of a patient condition improving is not equal to 0.5.

Step-by-step solution

Given information.

It is given that in a sample of 7000 asthma, ulcer, and herpes patients, 70% reported that an improvement by using the prescribed drug.

 Concept of testing of hypothesis

The null hypothesis, which can be further tested with the use of specific statistical tests, is an assumption that must be made before making any choice regarding the true value of the parameters. The choice is then made based on the calculated value of the parameter and critical, which means that if the computed value falls within the critical zone, the null hypothesis is rejected; otherwise, it is accepted.

Setting up the hypotheses

Under the claim the null and alternative hypotheses are,

Null hypothesis:

H₀ : p₀ = 0.5 ( 50 % )

The placebo is ineffective or the probability of a patient condition improving is 0.5.

Alternative hypothesis:

H₁ : p₀ ≠ 0.5 ( 50 % )

The placebo is ineffective or the probability of a patient condition improving is not equal to 0.5.

 Testing the hypothesis

Given level of significance is α = 0.05

To test the null hypothesis we have the test statistic z as

From the given information we have n=7000

p₀ = 0.5

Therefore,

And then

By substituting the values we get the test statistic value as,

From the standard normal table, the value of z_α for two tailed test and at the given level of significance α = 0.05 the critical value is z_0.05 = 1.96

Then , comparing the calculated value and table value of z

Thus, the null hypothesis is rejected and it is concluded that the placebo is ineffective; the probability of a patient condition improving is not equal to 0.5.