Give a $99.9 \%$ confidence interval, for $\mu_{1}-\mu_{2}$ given the following information.
\[
\begin{array}{l}
n_{1}=40, \bar{x}_{1}=2.43, s_{1}=0.95 \\
n_{2}=45, \bar{x}_{2}=2.89, s_{2}=0.64
\end{array}
\]

Step 1 ：Given the following values: \(n_{1}=40\), \(\bar{x}_{1}=2.43\), \(s_{1}=0.95\), \(n_{2}=45\), \(\bar{x}_{2}=2.89\), \(s_{2}=0.64\).

Step 2 ：We need to calculate the t-score for a 99.9% confidence interval with 39 degrees of freedom.

Step 3 ：Using the scipy.stats library in Python, we find the t-score to be approximately 3.558.

Step 4 ：Next, we calculate the standard error (se) using the formula \((s_{1}^{2}/n_{1} + s_{2}^{2}/n_{2})^{0.5}\), which gives us approximately 0.178.

Step 5 ：We then substitute the t-score and the given values into the formula to find the confidence interval. The lower bound of the confidence interval is calculated as \((\bar{x}_{1} - \bar{x}_{2}) - t_{score} * se\), which gives us approximately -1.093.

Step 6 ：The upper bound of the confidence interval is calculated as \((\bar{x}_{1} - \bar{x}_{2}) + t_{score} * se\), which gives us approximately 0.173.

Step 7 ：Thus, the 99.9% confidence interval for \(\mu_{1}-\mu_{2}\) is \(\boxed{(-1.093, 0.173)}\).