Step 1 :We are given the standard deviations of the two populations as \(\sigma_1 = 14.14\) and \(\sigma_2 = 10.88\), the sizes of the two samples as \(n_1 = 98\) and \(n_2 = 86\), and the confidence level as \(c = 0.85\).
Step 2 :We need to find the z-score corresponding to a confidence level of 0.85. This can be done using a standard normal distribution table or a z-score calculator. The z-score corresponding to a confidence level of 0.85 is approximately \(z_c = 1.4395314709384563\).
Step 3 :We can now calculate the margin of error using the formula: \[E = z_c \sqrt{\frac{{\sigma_1}^2}{n_1} + \frac{{\sigma_2}^2}{n_2}}\]
Step 4 :Substituting the given values into the formula, we get \[E = 1.4395314709384563 \sqrt{\frac{{14.14}^2}{98} + \frac{{10.88}^2}{86}}\]
Step 5 :Solving the above expression, we get \(E = 2.660855\)
Step 6 :Thus, the margin of error of the confidence interval for the difference between the two population means, rounded to six decimal places, is \(\boxed{2.660855}\)