Step 1 :State the hypotheses. The null hypothesis is that the proportions are equal, and the alternative hypothesis is that they are not equal. In mathematical terms, this is: \(H_{0}: p_{1}=p_{2}\) and \(H_{1}: p_{1} \neq p_{2}\)
Step 2 :Calculate the test statistic. The test statistic is given as \(z=1.14\)
Step 3 :Calculate the P-value. The P-value is given as \(P-value =0.252\)
Step 4 :Compare the P-value to the significance level. Since the P-value is greater than the significance level of \(\alpha=0.05\), we fail to reject the null hypothesis. This means that there is not sufficient evidence to support the claim that echinacea treatment has an effect.
Step 5 :Calculate the confidence interval for the difference in proportions. The formula for the confidence interval for the difference in proportions is \((p1 - p2) \pm z*\sqrt{(p1(1 - p1)/n1) + (p2(1 - p2)/n2)}\), where p1 and p2 are the sample proportions, n1 and n2 are the sample sizes, and z is the z-score for the desired confidence level (1.96 for a 95% confidence level).
Step 6 :Substitute the given values into the formula. The sample proportions are \(p1 = 0.898\) and \(p2 = 0.827\), the sample sizes are \(n1 = 49\) and \(n2 = 98\), and the z-score for a 95% confidence level is \(z = 1.96\).
Step 7 :Calculate the standard error (se). The standard error is \(se = 0.058\)
Step 8 :Calculate the lower and upper bounds of the confidence interval. The lower bound is \(ci_{lower} = -0.042\) and the upper bound is \(ci_{upper} = 0.185\)
Step 9 :State the final answer. The $95\%$ confidence interval for the difference in proportions is \(-0.042