Question

In a study testing the effects of placebos (sugar pills) on the headaches of 722 participants, when the participant believed that the pill was expensive, it was found that 299 participants had reduced pain after 30 minutes. We wish to test the claim that the actual proportion is less than 0.90 at the α=0.02 level of significance. What is the critical value?

If more than one critical value exists, please answer the the largest one.

Answer 1

The sample proportion here is computed as:
p = x/n = 299/722 = 0.4141

As we are testing the claim here whether the actual proportion is less than 0.90 at the α=0.02 level of significance, therefore we get from the standard normal tables here:

P(Z < -2.054) = 0.02

therefore -2.054 is the z critical value here.

The critical proportion value now is computed here as:

Therefore 0.8771 is the required critical proportion value here.