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confidence intervals, margin of error, alphas, and p-values

There's often a lot of confusion surrounding the terms confidence interval, margin of error, alphas, and p-values.

Let's begin with a confidence interval. The confidence interval formula gives you an interval that is likely to include your population value. A margin of error is really a part of the equation of confidence intervals in statistics. Here's the formula for confidence intervals:

Notice that the margin of error calculation actually includes a Z score. Moreover, the Z score is determined by the alpha (α) selected by the researcher. So while we're used to seeing the margin of error reported, it would be more accurate to also report the alpha selected by the researcher. Remember, the α determines the risk of being right/wrong and is selected by the researcher. You could still obtain a margin of error with a risk of being wrong 98% of the time! That's one of the reasons why it's important to report the α. When we see margin of errors reported in popular media, we usually assume that that researcher used one of the standard alphas (α = .05, .01, or .001), but that's an assumption on our part unless the researcher reports it. Keep in mind that we can report our alpha with phrases such as "at an α = .05..." or "at a confidence level of 95%..."

So what is the relationship between a p-value and an α? Well, the α determines the critical area and a p-value refers to the probability of obtaining a statistic beyond some defined critical area. That's why you'll often see it referred to in scientific papers as p > .05 or p < .05 (.05 is commonly used). We could write it like this:

An alpha value is a number 0≤α≤1 such that P(z≥z(observed) )≤α.

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So you could think of it as the α is the specific score while the p-value is the probability or area under the curve.

For further reading:


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