Success/failure condition in AP Statistics

In AP Statistics, the success/failure condition (also called the large counts condition) requires the number of successes and failures in each sample to be at least 10, so the sampling distribution of p̂ (or p̂₁ − p̂₂) is approximately normal and z-procedures for proportions are valid.

What is success/failure condition?

The success/failure condition is the normality check for inference about proportions. Proportion data is categorical, so you can't look at a histogram to see if it's "normal." Instead, you count. If a sample has enough successes AND enough failures, the sampling distribution of the sample proportion is approximately normal, and you're allowed to use a z-interval or z-test.

For a two-sample interval (Topic 3.10), the CED spells it out in 3.10.B.1. All four of these counts must be at least 10: n₁p̂₁, n₁(1−p̂₁), n₂p̂₂, n₂(1−p̂₂). That's successes and failures in sample 1, plus successes and failures in sample 2. Think of it as a "no lopsided samples" rule. If almost everyone said yes (or almost everyone said no), the distribution gets skewed and piles up against 0 or 1, and the normal model stops working.

Why success/failure condition matters in AP® Statistics

This term lives in Unit 3: Inference for Categorical Data: Proportions. It is spelled out in Topic 3.10, under learning objective AP Stats 3.10.B, which asks you to justify a confidence interval for the difference between two proportions by verifying conditions. Essential knowledge 3.10.B.1 names three conditions: randomization, the 10% condition (unless it's a randomized experiment), and normality, which is the success/failure check. It also matters for what comes after. The critical value z* in 3.10.C.2 and the margin of error in 3.10.D.2 only make sense if the sampling distribution is approximately normal. Skip this condition and the whole interval is built on a guess.

How success/failure condition connects across the course

10% Condition (Unit 3)

These two conditions travel together, but they protect different things. The success/failure condition protects the shape (normality). The 10% condition protects the standard error formula by making sure sampling without replacement doesn't break independence. On an FRQ, you check both, separately.

Standard Error (Unit 3)

The same p̂ values that go into your success/failure counts also go into the standard error, √(p̂₁(1−p̂₁)/n₁ + p̂₂(1−p̂₂)/n₂). Once the condition confirms the distribution is roughly normal, the SE tells you how spread out it is. Together they let you use z*.

One-Proportion z-Test (Unit 3)

The idea is the same, but the inputs change. In a confidence interval, you use the sample proportion p̂ because you don't know p. In a significance test, you assume the null is true, so you check np₀ and n(1−p₀) using the hypothesized value.

Normality for Means (Unit 4)

In Unit 4 the normality check looks totally different. For means, you use n ≥ 30 (the Central Limit Theorem) or show the sample data has no strong skew or outliers. Counting successes and failures only works for categorical data. For quantitative data, you look at the distribution itself.

Is success/failure condition on the AP® Statistics exam?

On multiple choice, expect stems like "An analyst confirms that the number of successes and failures in two independent samples are all greater than 10. What does satisfying this condition allow the analyst to assume?" The answer is that the sampling distribution of p̂₁ − p̂₂ is approximately normal. Common wrong choices say it guarantees independence, random sampling, or that the population itself is normal.

No released FRQ has used the phrase "success/failure condition" verbatim. Still, nearly every inference FRQ on proportions awards credit for checking conditions. To earn it, you need to:

• Show the actual numbers. Write out all four counts (for two samples) and state that each is at least 10. Saying "the condition is met" with no numbers usually won't earn credit.
• Use the right values. Use p̂ for intervals and p₀ for one-proportion tests.
• Say what it buys you. Tie it to the conclusion that the sampling distribution is approximately normal.

Success/failure condition vs 10% condition

Both conditions involve numbers and both show up in the same "check conditions" step, so they get mixed up constantly. The success/failure condition is about normality. It checks that np̂ and n(1−p̂) are each at least 10. The 10% condition is about independence. It checks that the sample is no more than 10% of the population when sampling without replacement. One quick memory trick is that the success/failure condition counts outcomes in your sample, while the 10% condition compares sample size to population size.

Key things to remember about success/failure condition

• The success/failure condition is how you check normality for inference about proportions.

• For a two-sample proportion interval, n₁p̂₁, n₁(1−p̂₁), n₂p̂₂, and n₂(1−p̂₂) must all be at least 10.

• Meeting the condition lets you treat the sampling distribution of p̂ (or p̂₁ − p̂₂) as approximately normal, which justifies using z*.

• Use sample proportions (p̂) for confidence intervals, but use the hypothesized value (p₀) when checking a one-proportion significance test.

• The success/failure condition and the 10% condition are separate checks: one protects normality and the other protects independence.

• On FRQs, show the actual counts and compare them to 10, because just writing "condition met" usually doesn't earn credit.

Frequently asked questions about success/failure condition

What is the success/failure condition in AP Stats?

It's the normality check for proportion inference. The number of successes and the number of failures in each sample must both be at least 10. When that holds, the sampling distribution of p̂ is approximately normal, so z-procedures are valid.

Does the success/failure condition mean the population is normal?

No. It says nothing about the population. Proportion data is categorical, so the population isn't "normal" at all. The condition only tells you the sampling distribution of p̂ (or p̂₁ − p̂₂) is approximately normal.

How is the success/failure condition different from the 10% condition?

The success/failure condition checks normality by requiring at least 10 successes and 10 failures. The 10% condition checks independence by requiring the sample to be no more than 10% of the population. Both use the number 10, but they answer different questions, and you need to check each one.

Do you use p̂ or p₀ for the success/failure condition?

It depends on the procedure. For a confidence interval, use p̂ because you don't know the true proportion. For a one-proportion z-test, use p₀ because you're assuming the null hypothesis is true.

How many checks do you need for a two-proportion confidence interval?

Four. Per CED essential knowledge 3.10.B.1, n₁p̂₁, n₁(1−p̂₁), n₂p̂₂, and n₂(1−p̂₂) must each be at least 10. If even one count falls below 10, the normal approximation isn't justified.