Answer to Question #178011 in Statistics and Probability for Calvin Lee

Question

A new drug cures 9 of 200 patients suffering from a type of cancer, for which the historical cure rate is 2%. Perform a test to check on the significance of this result, at both 5% and 1% levels of significance. Based on your conclusion, comments on the efficacy of the treatment using this new drug. 

Answer 1

$H_0 : p = 0.02 \\ H_1 : p ≠ 0.02 \\ n = 200 \\ \bar{p} = \frac{9}{200} = 0.045 \\ α_1 = 0.05$

Test statistic:

$Z = \frac{ \bar{p} -p}{\sqrt{ \frac{p(1-p)}{n}}} \\ Z = \frac{ 0.045 -0.02}{\sqrt{ \frac{(0.02)(0.08)}{200}}} = 2.53$

Critical value:

$Φ(Z_{cr}) = \frac{1 -α_1}{2} = 0.475 \\ Z_{cr}=1.96$

(−∞,−1.96) $\cup$ (1.96,∞) is the rejection region.

The test statistic Z falls into the rejection region. So we reject H₀ and accept H₁.

The efficacy of the treatment using this new drug is not equal to 2%.

$α_2 = 0.01 \\ Φ(Z_{cr}) = \frac{1 -α_2}{2} = 0.495 \\ Z_{cr}=2.58$

(−∞,−2.58) $\cup$ (2.58,∞) is the rejection region.

The test statistic Z does not fall into the rejection region. So we accept H_0.

The efficacy of the treatment using this new drug is equal to 2%.