First, find the sample proportion and its complement:
\[\hat{p} = \frac{x}{n} = \frac{112}{400} = 0.28 \qquad \hat{q} = 1 - \hat{p} = 0.72\]Verify the Central Limit Theorem applies:
Find the critical value. For a 95% confidence level, the remaining 5% is in the tails, with 2.5% in each tail. As we saw in Lesson 18.1 Critical Values, the critical value for a 95% confidence level is $z_c = \pm 1.96$.
Now calculate the margin of error:
\[\begin{align*} E &= z_c\sqrt{\frac{\hat{p}\hat{q}}{n}} \\ &= 1.96\sqrt{\frac{(0.28)(0.72)}{400}} \\ &= 1.96\sqrt{\frac{0.2016}{400}} \\ &= 1.96 \times 0.02245 \\ &\approx \mathbf{0.044} \end{align*}\]The margin of error is 0.044, or about 4.4 percentage points.