Confidence Intervals for a Large Sample

Question

The following $95\%$ confidence interval was constructed using a large sample of data: $(86.52,89.48)$. Which of the following could be a $99\%$ confidence interval for the same set of data?

$I. (86.98,89.02)$

$II. (86.37,89.63)$

$III. (87.04,88.98)$

My attempt: It is a large sample of data so we can approximate the sampling distribution with a Normal model. The mean is $$\bar{x} = \frac{86.52+89.48}{2}=88$$ The margin of error for the $95\%$ confidence interval is $$z^*\cdot (\text{Standard Error}) = 1.96(SE) = (89.48-88) = 1.48$$ This gives us that $SE$ is $.755$. The critical $z$ value for a $99\%$ interval is about $2.58$. The new margin of error is now $.755\cdot2.58 = 1.95$ So the $99\%$ confidence interval is now $(88-1.95,88+1.95) = (86.05,89.95)$. Which is not one of the answers. Where did I go wrong?

Answer 1

A higher confidence level will merely widen the interval. This leaves choice $II.$

Answer 2

It is the mean of any distribution that is asymptotically normal. Your 'large sample of data' isn't necessarily normal -- it could follow any distribution. Since a wider CI is always wider, it must be II.