# Finding a narrower confidence interval for a given CI, sample mean and size

I'm trying to understand confidence intervals but having some trouble. I've been doing some exercises I found online and I'm stuck on this question:

I have been given a 95% conﬁdence interval for a population proportion: (0.35, 0.40), a sample size of 200, and I need to find a 99% confidence interval. The methods I would go to first involve using standard deviation, which I don't have.

How can I approach this question without knowing variance? The whole quiz is about normal distributions, to give it some context.

edit: I realise this is essentially a homework question so I've added the tag.

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Hint: The (usual) formula for a confidence interval for a population proportion is different from the formulas for a confidence interval for a population mean. That formula involves the sample proportion, a confidence coefficient, and the sample size -- not a standard deviation (at least, not as a separate variable that you need to find the value of).

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Hint:

95% of the probability of a normal distribution is within $\pm 1.96$ standard deviations of the mean

99% of the probability of a normal distribution is within $\pm 2.576$ standard deviations of the mean

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Yep, I realise that. But how about for 99% of the means taken from random samples of the population? What I've read suggests it'll have student's t-distribution, but using that seems to require knowing variance. – Student Jun 16 '11 at 7:23
For a population of 200, using Student's $t$-distribution makes little difference. With 199 degrees of freedom, you have to scale by about $\frac{2.601}{1.972}$ rather than $\frac{2.576}{1.960}$, which is a small change if you are only reporting at two or three significant figures. – Henry Jun 16 '11 at 7:34
It's a sample of 200, the population is larger. Or am I missing your point? – Student Jun 16 '11 at 7:37
@Student: You have been implicitly told the standard deviation when you are told the width of a $95$ percent confidence interval. As to Student's distribution, Henry meant sample size. – André Nicolas Jun 16 '11 at 11:32
@Student: By the way, the title is kind of wrong. A $99$ percent confidence interval will be wider than a $95$ percent confidence interval. If you want to be more sure, you need to give a wider bound in order to catch the population proportion. – André Nicolas Jun 16 '11 at 11:42