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.