Answer to Question #130045 in Statistics and Probability for Ntokozo

Question

Suppose the Acme Drug Company develops a new drug, designed to prevent colds. The company
states that the drug is equally effective for men and women. To test this claim, they choose a
simple random sample of 100 women and 200 men from a population of 100,000 volunteers.
At the end of the study, 38% of the women caught a cold; and 51% of the men caught a cold.
Based on these findings, can we reject the company's claim that the drug is equally effective for
men and women? Use a 0.05 level of significance.

Answer 1

Let P₁: Population proportion of women who catch a cold and P₂: Population proportion of men who catch a cold after taking the drug.

Similarly, let $\hat{p_{1}}$ : Sample proportion of women who catch a cold and $\hat{p_{2}}$ : Sample proportion of men who catch a cold after taking the drug.

Since we have to test the claim that the drug is equally effective for men and women, we can set up the null and alternative hypothesis as, H₀ : P₁ = P₂ against H₁: P₁ not equal to P₂

Thus, we use a two-sample test for population proportions.

Under H₀, the test statistic is -

$Z_{calc} = \frac{\hat{p_{1}} - \hat{p_{2}}}{\sqrt{\hat{p}\hat{q}(\frac{1}{n_{1}}+\frac{1}{n_{2}})}} \sim N(0,1)$

n₁ and n₂ are the corresponding sample sizes.

And $\hat{p} = \frac{n_{1}\hat{p_{1}}+n_{2}\hat{p_{2}}}{n_{1}+n_{2}}$

We have $\hat{p_{1}} = 0.38, \hat{p_{2}} = 0.51, {n_{1}} =100, {n_{2}} = 200$

$\hat{p} = \frac{100*0.38 + 200*0.51}{100 + 200} = \frac{140}{300} = 0.46667$

Now,

$Z_{calc} = \frac{{0.38} - {0.51}}{\sqrt{{0.46667 * 0.5333}(\frac{1}{100}+\frac{1}{200})}} = \frac{-0.13}{0.06110} = -2.1276$

We have to test the claim at level of significance of 5% (Thus alpha = 0.05). Since its a two-tailed test, the critical value from standard normal tables is:

$Z_{\alpha/2} = Z_{0.025} = 1.96$

Test Criteria: Reject H₀ if |Z_calc| > Z_alpha/2 (i.e. 1.96)

Conclusion: Since our |Z_calc| = 2.1276 is greater than 1.96 we have sufficient evidence to reject H₀ at 5% level of significance.

Hence we can reject the company's claim that the drug is equally effective for men and women.