Researchers interested in determining the relative effectiveness of two different drug treatments on people with a chronic illness established two independent test groups. 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. Assume that the populations of times until remission for each of the two treatments are normally distributed with equal variance. Construct a $95 \%$ confidence interval for the difference $\mu_{1}-\mu_{2}$ between the mean number of days before remission after treatment $1\left(\mu_{1}\right)$ and the mean number of days before remission after treatment $2\left(\mu_{2}\right)$. Then find the lower limit and upper limit of the $95 \%$ confidence interval.

Carry your intermediate computations to at least three decimal places. Round your responses to at least two decimal places. (If necessary, consult a list of formulas.)
Lower limit:
Upper limit:

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 \(\boxed{3.42}\).