Find the margin of error for the given values of $c, s$, and $n$.
\[
c=0.90, s=3.4, n=8
\]
t-Distribution Table
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The margin of error is (Round to three decimal places as needed.)
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline & \begin{tabular}{c}
Level of \\
confidence, $c$
\end{tabular} & 0.80 & 0.90 & 0.95 & 0.98 & 0.99 & \\
\hline & One tail, $\alpha$ & 0.10 & 0.05 & 0.025 & 0.01 & 0.005 & \\
\hline d.f. & Two tails, $\alpha$ & 0.20 & 0.10 & 0.05 & 0.02 & 0.01 & d.f. \\
\hline 1 & & 3.078 & 6.314 & 12.706 & 31.821 & 63.657 & 1 \\
\hline 2 & & 1.886 & 2.920 & 4.303 & 6.965 & 9.925 & 2 \\
\hline 3 & & 1.638 & 2.353 & 3.182 & 4.541 & 5.841 & 3 \\
\hline 4 & & 1.533 & 2.132 & 2776 & 3.747 & 4.604 & 4 \\
\hline 5 & & 1.476 & 2.015 & 2.571 & 3.365 & 4.032 & 5 \\
\hline 6 & & 1.440 & 1943 & 2.447 & 3.143 & 3.707 & 6 \\
\hline 7 & & 1.415 & 1.895 & 2.365 & 2.998 & 3.499 & 7 \\
\hline 8 & & 1.397 & 1.860 & 2.306 & 2.896 & 3.355 & 8 \\
\hline 9 & & 1.383 & 1.833 & 2.262 & 2.821 & 3.250 & 9 \\
\hline 10 & & 1.372 & 1.812 & 2228 & 2.764 & 3.169 & 10 \\
\hline 11 & & 1.363 & 1.796 & 2.201 & 2.718 & 3.106 & 11 \\
\hline 12 & & 1.356 & 1.782 & 2.179 & 2.681 & 3.055 & 12 \\
\hline 13 & & 1350 & 1.771 & 2.160 & 2.650 & 3.012 & 13 \\
\hline 14 & & 1345 & 1.761 & 2.145 & 2.624 & 2.977 & 14 \\
\hline 15 & & 1341 & 1.753 & 2.131 & 2.602 & 2.947 & 15 \\
\hline 16 & & 1.337 & 1.746 & 2.120 & 2.583 & 2.921 & 16 \\
\hline 17 & & 1.333 & 1.740 & 2.110 & 2.567 & 2.898 & 17 \\
\hline 18 & & 1.330 & 1.734 & 2.101 & 2552 & 2.878 & 18 \\
\hline 19 & & 1.328 & 1.729 & 2.093 & 2.539 & 2.861 & 19 \\
\hline 20 & & 1.325 & 1.725 & 2.086 & 2.528 & 2.845 & 20 \\
\hline 21 & & 1.323 & 1.721 & 2.080 & 2.518 & 2.831 & 21 \\
\hline 22 & & 1.321 & 1,717 & 2.074 & 2.508 & 2819 & 22 \\
\hline 23 & & 1.319 & 1.714 & 2.069 & 2.500 & 2807 & 23 \\
\hline 24 & & 1.318 & 1.711 & 2.064 & 2.492 & 2.797 & 24 \\
\hline 25 & & 1.316 & 1.708 & 2.060 & 2.485 & 2.787 & 25 \\
\hline 26 & & 1.315 & 1.706 & 2.056 & 2.479 & 2.779 & 26 \\
\hline 27 & & 1.314 & 1.703 & 2.052 & 2.473 & 2.771 & 27 \\
\hline 28 & & 1.313 & 1.701 & 2.048 & 2.467 & 2.763 & 28 \\
\hline 29 & & 1.311 & 1.699 & 2.045 & 2.462 & 2.756 & 29 \\
\hline 30 & & 1.310 & 1.697 & 2.042 & 2.457 & 2.750 & 30 \\
\hline
\end{tabular}
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Answer
Final Answer: The margin of error for the given values of $c, s$, and $n$ is \(\boxed{2.278}\).
Step 1 ：Given that the level of confidence $c=0.90$, the standard deviation $s=3.4$, and the sample size $n=8$, we need to find the margin of error.
Step 2 ：First, we need to find the degrees of freedom, which is calculated as $n-1$. So, the degrees of freedom is $8-1=7$.
Step 3 ：Next, we look up the t-score in the t-distribution table for a level of confidence of $0.90$ and degrees of freedom of $7$. The t-score is $1.895$.
Step 4 ：We then substitute these values into the formula for the margin of error: $E = t \cdot \frac{s}{\sqrt{n}}$
Step 5 ：Substituting the given values, we get $E = 1.895 \cdot \frac{3.4}{\sqrt{8}}$
Step 6 ：Calculating the above expression, we find that the margin of error $E$ is approximately $2.278$.
Step 7 ：Final Answer: The margin of error for the given values of $c, s$, and $n$ is \(\boxed{2.278}\).
