In significance tests for a population proportion, we cannot blindly apply formulas; we must first verify key conditions so our inferences are valid and trustworthy.

Verifying Conditions for Statistical Inferences

When conducting a one-sample z-test for a population proportion, you are using a normal-model approximation to describe the sampling distribution of the sample proportion $\hat p$. This approximation is not automatically valid for every data set. The AP syllabus emphasizes two essential checks before proceeding with inference: the independence condition and the normality condition. Together, these conditions justify using a z-test and interpreting the resulting p-value as meaningful evidence about the population proportion.

Independence Condition: Random Process and the 10% Guideline

The independence condition focuses on how the data were produced. Statistical formulas for standard error and null distributions assume that each observation in the sample behaves as if it does not influence any other observation.

This figure shows a population of individuals, with 10 randomly selected subjects highlighted in a different color to represent a simple random sample. It emphasizes that every member of the population had an equal chance to be chosen, supporting the independence condition used in one-sample z-tests for a proportion. The health-research context in the surrounding text is extra, but the image itself focuses solely on the basic idea of simple random sampling. Source.

In practice, we do not usually check independence by examining individuals one by one. Instead, we examine the data collection method to decide whether independence is a reasonable assumption. The AP syllabus highlights two key ideas:

• Random process

  • Data should come from random sampling (from a defined population) or a randomized experiment (participants randomly assigned to treatment conditions).

  • Randomization helps break systematic links between observations, making independence plausible.

• 10% condition for sampling without replacement

  • When you sample without replacement from a finite population, selecting one individual slightly changes the population for the next selection, creating some dependence.

  • To keep this dependence small, require the sample size to be no more than 10% of the population size.

  • In words, the condition is: “The sample size is at most 10% of the population size,” which makes the dependence negligible for inference.

In AP Statistics, you justify the independence condition by clearly describing the random process and stating whether the 10% guideline is met when sampling without replacement. This written justification is an essential part of a complete inference solution.

Normality Condition: Expected Counts Under the Null

The normality condition addresses the shape of the sampling distribution of the sample proportion. For a one-sample z-test for a proportion, we rely on the sampling distribution of $\hat p$ being approximately normal if the null hypothesis is true. The syllabus states that the expected number of successes and failures under the null must both be at least 10.

To formalize this requirement, we work with expected counts under the null hypothesis value $p_0$.

EQUATION

For the normality condition, you check both of the following using the null value $p_0$ (not the sample proportion $\hat p$):

• Expected successes: $np_0 \ge 10$

• Expected failures: $n(1 - p_0) \ge 10$

When both inequalities are satisfied, the Central Limit Theorem and binomial approximation results suggest that the distribution of $\hat p$ under the null is close enough to normal for the z-test to be reliable.

This figure displays three histograms of the sampling distribution of $\hat p$ for the same population proportion but sample sizes n=100, n=300, and n=1000, each with an overlaid normal curve. As the sample size grows, the distributions become more bell-shaped and less spread out, mirroring the effect of meeting the conditions $np_0 \ge 10$ and $n(1 - p_0) \ge 10$. The specific values shown are illustrative and extend slightly beyond syllabus requirements, but they clearly demonstrate how increased sample size improves the normal approximation. Source.

If one or both conditions fail, the sampling distribution may be significantly skewed or discrete, making the usual z-test and its p-value less trustworthy.

Connecting Conditions to the One-Sample z-Test for a Proportion

The conditions you verify directly support the assumptions behind the z-test statistic and its null distribution:

• The independence condition justifies treating the sample proportion as having a stable, predictable variability. When observations are independent, the variance of $\hat p$ is well approximated by $p_0(1 - p_0)/n$, which underlies the standard error used in the test statistic.

• The normality condition justifies modeling the distribution of $\hat p$ (or the corresponding z-statistic) by a standard normal distribution when the null hypothesis is true. This is what allows you to translate the z-statistic into a p-value using the normal curve.

Because of this, verifying conditions is not a formality. It is the logical bridge between the real-world sampling process and the mathematical model used for inference.

How to Write Condition Checks in AP Responses

In AP-level work, you should clearly and concisely state your condition checks before computing the test statistic or p-value. A typical structure includes:

• Naming each condition and explicitly indicating whether it is met.

• Referencing the context of the problem when discussing the random process and population.

• Using the null hypothesis value $p_0$ when calculating expected successes and failures.

A well-written verification might include bullet points such as:

• Independence

  • The data come from a random sample (or randomized experiment) described in the problem.

  • The sample size is less than 10% of the population, so sampling without replacement does not seriously violate independence.

• Normality

  • Under $H_0$, both $np_0$ and $n(1 - p_0)$ are at least 10, so the sampling distribution of $\hat p$ is approximately normal.

By systematically verifying these conditions, you ensure that any z-test for a population proportion you perform is supported by the AP syllabus requirements for valid statistical inference.