A clinical trial is conducted comparing a new pain reliever for arthritis to a placebo. Participants are randomly assigned to receive the new treatment or a placebo and the outcome is pain relief within 30 minutes. The data are shown below. Is there a significant difference in the proportions of patients reporting pain relief? Run the test at a 5% level of significance.

. Pain Relief No Pain Relief
New medication 44 76
Placebo 21 99

Solution

Given that $n_1 = 120$, $X_1= 44$ number of participants from new medication group who get pain relief, $n_2=120$ and $X_2=21$ number of participants from placebo group who get pain relief.

The sample proportions are
$\hat{p}_1=\frac{X_1}{n_1}=\frac{44}{120}=0.367$.

$\hat{p}_2=\frac{X_2}{n_2}=\frac{21}{120}=0.175$.

The pooled estimate of sample proportion is
$\hat{p} =\frac{X_1+X_2}{n_1+n_2}=\frac{44+21}{120+120} =0.271$.

Step 1 State the hypothesis testing problem

The hypothesis testing problem is

$H_0 : p_1 = p_2$ against $H_1 : p_1 \neq p_2$ ($\textit{two-tailed}$)

Step 2 Define test statistic

The test statistic for testing above hypothesis testing problem is

$$ \begin{aligned} Z & =\frac{(\hat{p}_1-\hat{p}_2)-(p_1-p_2)}{\sqrt{\frac{\hat{p}(1-\hat{p})}{n_1}+\frac{\hat{p}(1-\hat{p})}{n_2}}}. \end{aligned} $$

The test statistic $Z$ follows standard normal distribution $N(0,1)$.

Step 3 Specify the level of significance $\alpha$

The significance level is $\alpha = 0.05$.

Step 4 Determine the critical value

As the alternative hypothesis is $\textit{two-tailed}$, the critical value of $Z$ $\text{ are }$ $\text{-1.96 and 1.96}$ (From Normal Statistical Table).

Z-critical region two-tailed
Z-critical region two-tailed

The rejection region (i.e. critical region) is $\text{Z < -1.96 or Z > 1.96}$.

Step 5 Computation

The test statistic under the null hypothesis is

$$ \begin{aligned} Z_{obs}&= \frac{(\hat{p}_1-\hat{p}_2)-(p_1-p_2)}{\sqrt{\frac{\hat{p}(1-\hat{p})}{n_1}+\frac{\hat{p}(1-\hat{p})}{n_2}}}\\ &= \frac{(0.367-0.175)-0}{\sqrt{\frac{0.271*(1-0.271)}{120}+\frac{0.271*(1-0.271)}{120}}}\\ &= 3.341 \end{aligned} $$

Step 6 Decision

Traditional approach:

The rejection region (i.e. critical region) is $\text{Z < -1.96 or Z > 1.96}$. The test statistic is $Z_{obs} =3.341$ which falls $inside$ the critical region, we $\textit{reject}$ the null hypothesis.

OR

$p$-value approach:

The test is $\text{two-tailed}$ test, so the p-value is the area to the $\text{extreme}$ of the test statistic ($Z_{obs}=3.341$) is p-value = $0.0008$.

The p-value is $0.0008$ which is $\textit{less than}$ the significance level of $\alpha = 0.05$, we $\textit{reject}$ the null hypothesis.

That is our data support the claim that there is a significant difference in the proportions of patients reporting pain relief.