Step 1 :Given that the test statistic is \(z=0.70\) and the P-value is \(0.483\).
Step 2 :Since the P-value is greater than the significance level of \(\alpha=0.05\), we fail to reject the null hypothesis. There is not sufficient evidence to support the claim that echinacea treatment has an effect.
Step 3 :To test the claim by constructing an appropriate confidence interval, we need to calculate the difference in proportions \(p1 - p2\) and then construct a 95% confidence interval around this difference.
Step 4 :The proportions \(p1\) and \(p2\) are the proportions of subjects who developed minovirus infections in the echinacea and placebo groups, respectively. These can be calculated as follows: \(p1 = 43 / 49\) and \(p2 = 80 / 96\).
Step 5 :The difference in proportions is then \(p1 - p2\).
Step 6 :The standard error for the difference in proportions can be calculated using the formula: \(SE = \sqrt{(p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2)}\) where \(n1\) and \(n2\) are the sizes of the echinacea and placebo groups, respectively.
Step 7 :The 95% confidence interval can then be calculated as follows: \(CI = (p1 - p2) ± (z * SE)\) where \(z\) is the z-score for a 95% confidence interval, which is approximately 1.96.
Step 8 :Calculating these values, we get \(p1 = 0.878\), \(p2 = 0.833\), \(diff = 0.044\), \(SE = 0.060\), \(CI_{lower} = -0.074\), and \(CI_{upper} = 0.162\).
Step 9 :\(\boxed{\text{Final Answer: The 95% confidence interval is } -0.074