Margin of Error and Confidence Intervals

I'm trying to understand the difference between a margin of error and a confidence interval with p-value .05. The margin of error gives a range of values around a sample mean such that the population mean has a 95% chance of being within that range. Isn't this the same thing as the meaning of a confidence interval? My book says otherwise but doesn't make it clear how that's so.

Re: Margin of Error and Confidence Intervals

Also, I want to make sure I'm not being stupid about something. I keep seeing this kind of language in my stats books and it seems wrong. For instance, they'll say, for the sampling distribution of the difference of two sample means, that the mean of $\displaystyle \overline{x}_{1} - \overline{x}_{2}$ is $\displaystyle \mu_{1} - \mu_{2}$. Presumably this means, if the sample size is large enough, these are approximately equal, no?

But things get worse. Then they say, "If the sampled populations are normally distributed, then the sampling distribution of $\displaystyle \overline{x}_{1} - \overline{x}_{2}$ is **exactly** normally distributed, regardless of the sample size." Really? Even if $\displaystyle n_{1} = n_{2} = 1$? Even if you just take one such sample? Or two?

Are they just being imprecise in their choice of words, even when they're emphasizing that their claim is to be taken as precise? Or am I missing something?

Re: Margin of Error and Confidence Intervals

I think wikipedia does a good job explaining it.

Quote:

The margin of error is usually defined as the "radius" (or half the width) of a confidence interval for a particular statistic from a survey.