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### Course: AP®︎/College Statistics>Unit 10

Lesson 8: Confidence intervals for the difference of two proportions

# Calculating a confidence interval for the difference of proportions

Calculating two-sample z interval to estimate the difference between two population proportions.

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• What happens when sample sizes are small? Just like single proportions case, we use t distribution?
• We could do a randomization test (also called a permutation test), but in general, it's just odd to use small samples to estimate proportions. Percentages/proportions try to place values on a scale of 0-1 (or 0% to 100%), so we don't get anywhere near the precision we're looking for when we use a small sample.
(1 vote)
• Why do we not use a pooled proportion for the standard error here but we use it when we are looking calculating a p-value? Many times it seems that we evaluate significance based on whether the confidence interval crosses 0 to determine significance which often relates to the p-value. That is why I am confused why we don't use the pooled proportion here
• From the author:Hi! Here, we're making a confidence interval. The goal is to estimate the difference between the true underlying population proportions Pn and Ps. There's no assumption that those proportions are the same — we just want to estimate how different they might be.

A significance test has a different goal and set of assumptions. To test IF there's a difference, we assume that there is no difference between Ps and Pn. Then, we look at the sample difference and see if it could reasonably happen by chance alone when Pn and Ps are equal. We pool the proportions to get an estimate of that common value to be consistent with our assumption of equality in the null hypothesis.

Note that neither method is perfect for standard error, but they key is that they both work pretty well as advertised when we meet all of the conditions (eg a 95% CI will capture the true difference about 95% of the time, and a test with alpha = 0.05 will reject/fail to reject the null hypothesis about as often as it's supposed to.
• Suppose we have independent random samples of size n1=615 and n2=605. The proportions of success in the two samples are p1=.53 and p2=.45. Find the 90% confidence interval for the difference in the two population proportions
(1 vote)
• If the difference in the proportion is p1-p2 then the confidence interval would be (0.033, 0.127).
• If the problem doesn't specify (p sub s - p sub n) like this problem does, does it matter which value should be subtracted from the other in the first term? Should that first term in the equation be nonnegative?