How do we use the p-value to decide if there is statistically significant evidence that the coin is unfair — or that our new product experience is an improvement on the status quo? It comes back to that 5% false positive rate that we agreed to accept at the beginning: we conclude that there is a statistically significant effect if the p-value is less than 0.05. This formalizes the intuition that we should reject the null hypothesis that the coin is fair if our result is sufficiently unlikely to occur under the assumption of a fair coin. In the example of observing 55 heads in 100 coin flips, we calculated a p-value of 0.32. Because the p-value is larger than the 0.05 significance level, we conclude that there is not statistically significant evidence that the coin is unfair.

There are two conclusions that we can make from an experiment or A/B test: we either conclude there is an effect (“the coin is unfair”, “the Top 10 feature increases member satisfaction”) *or* we conclude that there is insufficient evidence to conclude there is an effect (“cannot conclude the coin is unfair,” “cannot conclude that the Top 10 row increases member satisfaction”). It’s a lot like a jury trial, where the two possible outcomes are “guilty” or “not guilty” — and “not guilty” is very different from “innocent.” Likewise, this (frequentist) approach to A/B testing does not allow us to make the conclusion that there is no effect — we never conclude the coin is fair, or that the new product feature has no impact on our members. We just conclude we’ve not collected enough evidence to reject the null assumption that there is no difference. In the coin example above, we observed 55% heads in 100 flips, and concluded we had insufficient evidence to label the coin as unfair. Critically, we did not conclude that the coin was fair — after all, if we gathered more evidence, say by flipping that same coin 1000 times, we might find sufficiently compelling evidence to reject the null hypothesis of fairness.

## Rejection Regions and Confidence Intervals

There are two other concepts in A/B testing that are closely related to p-values: the **rejection region** for a test, and the **confidence interval** for an observation. We cover them both in this section, building on the coin example from above.

**Rejection Regions. **Another way to build a decision rule for a test is in terms of what’s called a “rejection region” — the set of values for which we’d conclude that the coin is unfair. To calculate the rejection region, we once more assume the null hypothesis is true (the coin is fair), and then define the rejection region as the set of least likely outcomes with probabilities that sum to no more than 0.05. The rejection region consists of the outcomes that are the most extreme, provided the null hypothesis is correct — the outcomes where the evidence against the null hypothesis is strongest. If an observation falls in the rejection region, we conclude that there is statistically significant evidence that the coin is not fair, and “reject” the null. In the case of the simple coin experiment, the rejection region corresponds to observing fewer than 40% or more than 60% heads (shown with blue shaded bars in Figure 3). We call the boundaries of the rejection region, here 40% and 60% heads, the critical values of the test.

There is an equivalence between the rejection region and the p-value, and both lead to the same decision: the p-value is less than 0.05 if and only if the observation lies in the rejection region.

**Confidence Intervals**. So far, we’ve approached building a decision rule by first starting with the null hypothesis, which is always a statement of no change or equivalence (“the coin is fair” or “the product innovation does not impact member satisfaction”). We then define possible outcomes under this null hypothesis and compare our observation to that distribution. To understand confidence intervals, it helps to flip the problem around to focus on the observation. We then go through a thought exercise: given the observation, what values of the null hypothesis would lead to a decision not to reject, assuming we specify a 5% false positive rate? For our coin flipping example, the observation is 55% heads in 100 flips and we do not reject the null of a fair coin. Nor would we reject the null hypothesis that the probability of heads was 47.5%, 50%, or 60%. There’s a whole range of values for which we would not reject the null, from about 45% to 65% probability of heads (Figure 4).

This range of values is a confidence interval: the set of values under the null hypothesis that would not result in a rejection, given the data from the test. Because we’ve mapped out the interval using tests at the 5% significance level, we’ve created a 95% confidence interval. The interpretation is that, under repeated experiments, the confidence intervals will cover the true value (here, the actual probability of heads) 95% of the time.

There is an equivalence between the confidence interval and the p-value, and both lead to the same decision: the 95% confidence interval does not cover the null value if and only if the p-value is less than 0.05, and in both cases we reject the null hypothesis of no effect.

## Summary

Using a series of thought exercises based on flipping coins, we’ve built up intuition about false positives, statistical significance and p-values, rejection regions, confidence intervals, and the two decisions we can make based on test data. These core concepts and intuition map directly to comparing treatment and control experiences in an A/B test. We define a “null hypothesis” of no difference: the “B” experience does not alter affect member satisfaction. We then play the same thought experiment: what are the possible outcomes and their associated probabilities for the difference in metric values between the treatment and control groups, *assuming there is no difference in member satisfaction*? We can then compare the observation from the experiment to this distribution, just like with the coin example, calculate a p-value and make a conclusion about the test. And just like with the coin example, we can define rejection regions and calculate confidence intervals.

But false positives are only of the two mistakes we can make when acting on test results. In the next post in this series, we’ll cover the other type of mistake, false negatives, and the closely related concept of statistical power. Follow the Netflix Tech Blog to stay up to date.