A light bulb manufacturer wants to compare the mean lifetimes of two of its light bulbs, model $A$ and model $B$. Independent random samples of the two models were taken. Analysis of 13 bulbs of model A showed a mean lifetime of 1314 hours and a standard deviation of 85 hours. Analysis of 14 bulbs of model B showed a mean lifetime of 1306 hours and a standard deviation of 92 hours. Assume that the populations of lifetimes for each model are normally distributed and that the variances of these populations are equal. Construct a $90 \%$ confidence interval for the difference $\mu_{1}-\mu_{2}$ between the mean lifetime $\mu_{1}$ of model A bulbs and the mean lifetime $\mu_{2}$ of model $B$ bulbs. Then find the lower limit and upper limit of the $90 \%$ 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.)

Step 1 ：Given in the problem, we have: \(\bar{x}_1 = 1314\) (mean of model A), \(s_1 = 85\) (standard deviation of model A), \(n_1 = 13\) (sample size of model A), \(\bar{x}_2 = 1306\) (mean of model B), \(s_2 = 92\) (standard deviation of model B), \(n_2 = 14\) (sample size of model B).

Step 2 ：The t-score for a 90% confidence level with degrees of freedom \(df = n_1 + n_2 - 2 = 13 + 14 - 2 = 25\) can be found in a t-distribution table or calculated using a statistical software or calculator. The t-score for a 90% confidence level (two-tailed) and 25 degrees of freedom is approximately 1.708.

Step 3 ：Substituting these values into the formula, we get: \(CI = (\bar{x}_1 - \bar{x}_2) \pm t\sqrt{(s_1^2/n_1) + (s_2^2/n_2)} = (1314 - 1306) \pm 1.708\sqrt{(85^2/13) + (92^2/14)} = 8 \pm 1.708\sqrt{(7225/13) + (8464/14)} = 8 \pm 1.708\sqrt{556.154 + 604.571} = 8 \pm 1.708\sqrt{1160.725} = 8 \pm 1.708*34.079 = 8 \pm 58.215\)

Step 4 ：So, the 90% confidence interval for the difference between the mean lifetimes of model A and model B light bulbs is approximately (-50.22 hours, 66.22 hours). This means we are 90% confident that the true difference in mean lifetimes between model A and model B light bulbs is between -50.22 hours and 66.22 hours.

Step 5 ：Final Answer: \(\boxed{(-50.22, 66.22)}\)