Question

Past data indicates an 80% (0.8) success rate in treating a certain medical problem. Researchers want to test if a new treatment will increase the success rate above 80%. This new treatment was used on 100 patients in a clinical trial and was successful on 87 of those patients. Write the null and alternative hypothesis and test at 5% significance level.

Answer 1

Null hypothesis $H_0:\;p=0.8.$

Alternative hypothesis $H_a:\;p>0.8.$

Test statistic: $z=\frac{\hat p-p}{\sqrt{\frac{p(1-p)}{n}}}=\frac{0.87-0.8}{\sqrt{\frac{0.8(1-0.8)}{100}}}=1.75.$

P-value: $p=P(Z>1.75)=0.0401.$

Since the P-value is less than 0.05, reject the null hypothesis.