P-Value and Test Statistic

Question

Physicians at a clinic gave what they thought were drugs to 840 patients. Although the doctors later learned that the drugs were really placebos, 58% of the patients reported an improved condition. Assume that if the placebo is ineffective, the probability of a patients condition improving is 0.55. Test the hypotheses that the proportion of patients improving is >0.55. 

Find the test statistics: 

z=  

Find the p-value.

𝑝=

Answer 1

Ho: p=0.55

H1: p>0.55

Type I error = alpha = .05 (usual value)

Standard error under Ho = square root of (po * qo/n) where po = 0.55 and qo = 1-po = 0.45

So, the standard error = square root of (.55*.45/840) = 0.017165

z = (sample proportion - po)/standard error = (.58-.55)/0.017165 = 1.7477

The p-value is Prob(z>1.7477 assuming Ho is true) = 0.0403

Reject Ho.