Find the margin of error $E$ in finding the mean if $c=0.97$, $\sigma=3.8$, and $n=49$.

Step 1 ：We are given that the confidence level $c=0.97$, the standard deviation of the population $\sigma=3.8$, and the size of the sample $n=49$.

Step 2 ：We need to find the z-score $Z_c$ corresponding to the confidence level $c=0.97$. This can be found using a z-table or a statistical calculator. In this case, $Z_c = 2.17009037758456$.

Step 3 ：The margin of error in finding the mean can be calculated using the formula: $E = Z_c * \frac{\sigma}{\sqrt{n}}$.

Step 4 ：Substituting the given values into the formula, we get $E = 2.17009037758456 * \frac{3.8}{\sqrt{49}}$.

Step 5 ：Solving the above expression, we get $E = 1.1780490621173325$.

Step 6 ：Rounding to three decimal places, the margin of error $E$ in finding the mean is approximately \(\boxed{1.178}\).