Given two dependent random samples with the following results:
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline Population 1 & 19 & 39 & 35 & 22 & 35 & 32 & 37 \\
\hline Population 2 & 29 & 25 & 40 & 16 & 27 & 19 & 32 \\
\hline
\end{tabular}
Use this data to find the $98 \%$ confidence interval for the true difference between the population means. Assume that both populations are normally distributed.
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Step 3 of 4: Calculate the margin of error to be used in constructing the confidence interval. Round your answer to six decimal places.

Step 1 ：We are given two dependent random samples from two populations. The data from Population 1 is [19, 39, 35, 22, 35, 32, 37] and from Population 2 is [29, 25, 40, 16, 27, 19, 32].

Step 2 ：We need to find the 98% confidence interval for the true difference between the population means. To do this, we first need to calculate the standard deviation of the differences between the two populations.

Step 3 ：The differences between the two populations are [-10, 14, -5, 6, 8, 13, 5]. The standard deviation of these differences, denoted as \(s\), is approximately 8.92295061171178.

Step 4 ：The sample size, denoted as \(n\), is 7 since there are 7 data points in each population.

Step 5 ：The critical value for a 98% confidence level, denoted as \(Z_{\alpha/2}\), is approximately 2.33. This value can be found in a Z-table or using a statistical calculator.

Step 6 ：We can now calculate the margin of error, denoted as \(E\), using the formula \(E = Z_{\alpha/2} \times \frac{s}{\sqrt{n}}\). Substituting the values we have, \(E = 2.33 \times \frac{8.92295061171178}{\sqrt{7}}\), we get \(E\) to be approximately 7.8457438909383415.

Step 7 ：Thus, the margin of error to be used in constructing the 98% confidence interval for the true difference between the population means is approximately \(\boxed{7.845743}\).