For this sub-competency, we will be using confidence intervals and hypothesis testing as tools for comparing *two* populations. Since the goal of most experiments is to compare two (or more) treatments, we need rigorous statistical tools to identify when there are significant differences between the treatments. Do NOT freak out about the size and messiness of the formulas. On the whole, they are the same as we’ve used before, at least in terms of the *language* of the terms used. However, because we’re dealing with two (or more) populations now, most formulas are going to be scaled-up to include values from both populations. There are going to be plenty of subscripts!

One of the most important ideas to keep in mind is this: if two things are equal, then the difference between them is zero. In symbols, if *a* = *b*, then *a* – *b* = 0. That should be pretty obvious. But when we jump to comparing, for example, the mean of two different populations (*μ*_{1} and *μ*_{2}), the notation can start to confuse the best of us. For example:

if *μ*_{1} = *μ*_{2}, then:

*μ*_{1} – *μ*_{2} = 0

The number 0 is going to be very important to us. Because of this, we’ll have the two following goals:

- When creating a confidence interval about the
*difference of two populations*, we’re going to be solely interested in whether our interval contains the number 0. - When conducting a hypothesis test, our null hypothesis (which tests the condition of
*no difference between the two populations*) will be:*H*_{0}:*μ*_{d}= 0, where*μ*_{d}stands for the difference in the two populations, i.e.,*μ*_{d}=*μ*_{1}–*μ*_{2}. Our alternative hypothesis will be that*μ*_{d}is either greater than, less than, or not equal to (different from) zero.