According to the historical data, the life expectancy in Germany is less than the life expectancy in the United States. A new study has been made to see whether this has changed. Records of 225 individuals from Germany who died recently are selected at random. The 225 individuals lived a mean of 77.9 years with a standard deviation of 5.8 years. Records of 205 individuals from the United States who died recently are selected at random and independently. The 205 individuals lived a mean of 76.6 years with a standard deviation of 6.3 years. Assume that the population standard deviations of lifetimes in these two countries can be estimated by the sample standard deviations, as the samples used were quite large. Construct a $95 \%$ confidence interval for $\mu_{1}-\mu_{2}$, the difference between the life expectancy $\mu_{1}$ in Germany and the life expectancy $\mu_{2}$ in the United States. 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 answers to at least two decimal places. (If necessary, consult a list of formulas.)
\begin{tabular}{|l|l|}
\hline Lower limit: \\
Upper limit:
\end{tabular}$\quad \times \quad+\quad \times$

Step 1 ：Given that the sample mean life expectancy in Germany, \(\bar{x}_1\), is 77.9 years, the sample mean life expectancy in the United States, \(\bar{x}_2\), is 76.6 years, the sample standard deviation of life expectancy in Germany, \(s_1\), is 5.8 years, the sample standard deviation of life expectancy in the United States, \(s_2\), is 6.3 years, the sample size from Germany, \(n_1\), is 225, and the sample size from the United States, \(n_2\), is 205.

Step 2 ：We are asked to find a 95% confidence interval for the difference between the mean life expectancy in Germany and the United States. The z-score corresponding to a 95% confidence interval, \(z\), is 1.96.

Step 3 ：The formula for the confidence interval for the difference between two means is \((\bar{x}_1 - \bar{x}_2) \pm z \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\).

Step 4 ：Substituting the given values into the formula, we get \((77.9 - 76.6) \pm 1.96 \sqrt{\frac{5.8^2}{225} + \frac{6.3^2}{205}}\).

Step 5 ：Solving the above expression, we get the lower limit as 0.15 and the upper limit as 2.45.

Step 6 ：Thus, the 95% confidence interval for the difference between the life expectancy in Germany and the life expectancy in the United States is \(\boxed{(0.15, 2.45)}\) years. This means we are 95% confident that the true difference in life expectancy between Germany and the United States is between 0.15 years and 2.45 years, with Germany having a higher life expectancy.