Step 1 :Given that the first group consisted of 11 people with the illness, and the second group consisted of 10 people with the illness. The first group received treatment 1 and had a mean time until remission of 179 days with a standard deviation of 8 days. The second group received treatment 2 and had a mean time until remission of 182 days with a standard deviation of 6 days.
Step 2 :We are asked to construct a 95% confidence interval for the difference between the mean number of days before remission after treatment 1 and the mean number of days before remission after treatment 2.
Step 3 :The formula for a confidence interval for the difference between two means (assuming equal variances) is: \[(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2, n_1+n_2-2} \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\]
Step 4 :Where: \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means, \(s_1\) and \(s_2\) are the sample standard deviations, \(n_1\) and \(n_2\) are the sample sizes, \(t_{\alpha/2, n_1+n_2-2}\) is the t-score for a two-tailed test with \(\alpha = 0.05\) and degrees of freedom \(df = n_1 + n_2 - 2\).
Step 5 :Substituting the given values into the formula, we get: \[t_{\alpha/2, n_1+n_2-2} = 2.093024054408263\], \[\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = 3.0689056385268376\]
Step 6 :Calculating the lower limit of the confidence interval, we get: \[-9.42329332214582\]
Step 7 :Calculating the upper limit of the confidence interval, we get: \[3.4232933221458213\]
Step 8 :Rounding to two decimal places, the lower limit of the 95% confidence interval for the difference between the mean number of days before remission after treatment 1 and the mean number of days before remission after treatment 2 is \(\boxed{-9.42}\) and the upper limit is \(\boxed{3.42}\).