In AP Statistics, the null distribution is the probability distribution of the test statistic assuming the null hypothesis is true. It can be a randomization distribution built by simulation or a theoretical distribution like the z, t, or chi-square distribution, and it is what you use to find p-values.
The null distribution answers one question. If the null hypothesis were actually true, what values would my test statistic take, and how often? Every significance test in AP Stats works by comparing your observed test statistic to this "world where H₀ is true," and the p-value is just the proportion of that null distribution as extreme or more extreme than what you got.
The CED says the null distribution can come in two flavors. It can be a randomization distribution, where you simulate tons of samples assuming the null parameter value and collect the resulting statistics. Or it can be a theoretical distribution when a probability model fits, which is what you'll use almost every time on the exam. For a one-proportion z-test it's the standard normal (z) distribution, for a slope test it's a t-distribution with n-2 degrees of freedom, and for a chi-square goodness-of-fit test it's a chi-square distribution with (number of categories - 1) degrees of freedom. Different test, different shape, same job.
This idea is the connective tissue of the entire inference half of the course. It shows up explicitly in the essential knowledge for 6.5.A (the null distribution for a proportion test is a randomization distribution or a z distribution), 8.3.A (the null distribution for a goodness-of-fit statistic is chi-square), and 9.5.A (the null distribution for a slope test statistic is a t-distribution with df = n-2). It also traces back to Topic 5.3, where sampling distributions and randomization distributions are first built by simulation. When 6.5.B asks you to interpret a p-value as "the proportion of values for the null distribution that are as extreme or more extreme than the observed test statistic," you literally cannot write a correct interpretation without this concept. If you understand the null distribution, p-values stop being a magic number from your calculator and start making sense.
Keep studying AP® Statistics Unit 3
Sampling Distributions and the CLT (Unit 5)
A null distribution is just a sampling distribution with a specific assumption baked in, namely that the parameter equals the value in H₀. The CLT is why so many null distributions end up approximately normal when n is large.
Interpreting p-Values (Unit 6)
The p-value is defined directly from the null distribution. It's the area in the tail(s) at or beyond your observed test statistic, with which tail depending on whether the alternative is >, <, or ≠.
Chi-Square Statistic (Unit 8)
The chi-square statistic only takes non-negative values, so its null distribution is right-skewed instead of bell-shaped, and the p-value always comes from the right tail. Same logic as a z-test, different shape.
Degrees of Freedom (Units 8-9)
Degrees of freedom tell you exactly which t or chi-square curve is your null distribution. For a slope test it's n-2; for goodness of fit it's categories minus 1. Get df wrong and you're reading the p-value off the wrong curve.
Multiple-choice questions love asking you to identify the correct null distribution for a given test. A classic stem gives regression output like b = 2.5, s_b = 0.8, n = 15 and asks for both the test statistic and its null distribution (here, t with 13 degrees of freedom). Another version states H₀: β = 2 with 40 data points and asks for the appropriate null distribution of (b - 2)/SE_b, which is t with df = 38. Chi-square questions test whether you know the null distribution is chi-square with the right df and when that approximation is valid. On FRQs, the term shows up in your p-value interpretation. Full credit requires saying the p-value assumes the null hypothesis is true, which is exactly the null distribution assumption in words. Naming the wrong distribution or the wrong df is one of the most common ways to lose points on inference FRQs.
A sampling distribution describes how a statistic varies across all possible samples, with no assumption about what the parameter equals. A null distribution is the special case where you assume the parameter equals the null value (like p = p₀ or β = 0). Every null distribution is a sampling distribution, but built under a specific "H₀ is true" assumption. That's why the standard error in a one-proportion z-test uses p₀, not p̂.
The null distribution is the distribution of the test statistic assuming the null hypothesis is true, and it's where every p-value comes from.
It can be a randomization distribution generated by simulation or a theoretical distribution like z, t, or chi-square when conditions are met.
Each test has its own null distribution. Proportions use z, regression slope tests use t with n-2 degrees of freedom, and goodness-of-fit tests use chi-square with (categories - 1) degrees of freedom.
The p-value is the proportion of the null distribution as extreme or more extreme than your observed test statistic, with the relevant tail set by the alternative hypothesis.
Any p-value interpretation must state that it was computed assuming the null hypothesis is true, which is what makes it a null distribution in the first place.
It's the probability distribution of the test statistic assuming the null hypothesis is true. You compare your observed statistic to it to find the p-value, the proportion of values as extreme or more extreme than what you observed.
Not quite. A sampling distribution makes no assumption about the parameter, while a null distribution assumes the parameter equals the null value (like p = p₀). A null distribution is a sampling distribution built under the H₀-is-true assumption.
No. It's approximately normal (z) for proportion tests, but it's a t-distribution for slope tests (df = n-2) and a right-skewed chi-square distribution for goodness-of-fit tests (df = categories - 1). The test determines the shape.
When all conditions are satisfied, the test statistic t = (b - β₀)/SE_b follows a t-distribution with n-2 degrees of freedom. So with n = 15 data points, you'd use a t-distribution with 13 degrees of freedom.
The null hypothesis is a claim about a parameter, like H₀: p = 0.5. The null distribution is what that claim implies about your test statistic's behavior across all possible samples. The hypothesis is the assumption; the distribution is its consequence.
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