Given two dependent random samples with the following results:
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline Population 1 & 42 & 49 & 31 & 29 & 34 & 48 & 27 \\
\hline Population 2 & 46 & 40 & 27 & 36 & 19 & 41 & 33 \\
\hline
\end{tabular}

Use this data to find the $80 \%$ 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.
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Step 1 ：Given two dependent random samples from Population 1 and Population 2 as follows: Population 1: [42, 49, 31, 29, 34, 48, 27] and Population 2: [46, 40, 27, 36, 19, 41, 33].

Step 2 ：Calculate the differences between the two populations to get: [-4, 9, 4, -7, 15, 7, -6].

Step 3 ：Calculate the standard deviation of the differences, which is approximately \(8.423323628614833\).

Step 4 ：The sample size (n) is 7, and the degrees of freedom (df) is 6.

Step 5 ：Find the critical value for an 80% confidence interval from the t-distribution, which is approximately \(1.4397557472577693\).

Step 6 ：Calculate the margin of error, which is the critical value times the standard deviation divided by the square root of the sample size. The margin of error is approximately \(4.583774958210464\).

Step 7 ：The margin of error to be used in constructing the 80% confidence interval is approximately \(\boxed{4.583775}\).