Students must verify that the sampling distribution of the sample proportion is approximately normal when performing a one-sample z-test for a population proportion.

Understanding the Purpose of Normality Checks

Checking normality for inference ensures that the test statistic used in a one-sample z-test behaves according to the standard normal distribution. This requirement matters because the z-test relies on approximating the distribution of the sample proportion with a normal model. When this approximation is justified, probability calculations and p-values accurately reflect the behavior of repeated samples under the null hypothesis.

The Role of the Null Hypothesis in Normality Assessment

Normality is assessed under the assumption that the null hypothesis is true, meaning the population proportion equals a specified value, p₀. Because inference is anchored to this assumption, expected counts of successes and failures must be evaluated using p₀, not the sample proportion. This aligns the model with the hypothesized distribution forming the basis of the test.

Conditions for Approximate Normality

The sampling distribution of the sample proportion is considered approximately normal when two fundamental criteria are met:

Minimum Successes and Failures

To justify normality, both expected successes and expected failures must reach a sufficient minimum. This ensures that the binomial distribution underlying the sample proportion is well-approximated by a normal model.

• Expected successes: np₀ ≥ 10

• Expected failures: n(1 – p₀) ≥ 10

These thresholds guarantee that the distribution is not excessively skewed and that the z-test yields reliable probability estimates.

This figure compares a binomial distribution to its normal approximation. The bars represent binomial probabilities, while the curve shows the normal density used as an approximation. It includes minor extra detail by displaying counts of successes rather than proportions, though the concept underlying approximate normality is the same. Source.

EQUATION

Because these values depend entirely on the null hypothesis, they remain fixed for any given test design. This provides a consistent framework for assessing whether the z-test can be legitimately applied.

Why Expected Counts Matter

Expected successes and failures reflect how often each outcome is anticipated if the null hypothesis were true. When these expected counts are too small, the distribution of the sample proportion becomes noticeably non-normal. This undermines the assumptions behind both the z-statistic and its corresponding p-value. When counts meet the required minimums, however, sampling variability behaves predictably, allowing the normal model to serve as a reasonable approximation.

Evaluating Whether the Conditions Are Met

Students should systematically evaluate the normality condition using a structured approach:

Step-by-Step Normality Check

• Identify the sample size, n.

• Identify the hypothesized population proportion, p₀, from the null hypothesis.

• Compute the expected number of successes using np₀.

• Compute the expected number of failures using n(1 − p₀).

• Compare the results to the required minimum thresholds of 10 for each category.

If either expected count falls below 10, the normal approximation is not reliable, and the z-test should not be used.

Key Terminology Supporting Normality Assessment

When introducing the z-test, the term sampling distribution arises frequently.

Understanding this definition clarifies why the normal model is needed: it predicts the behavior of the statistic across repeated samples.

This figure illustrates simulated sampling distributions of the sample proportion for increasing sample sizes, each overlaid with a normal curve. It demonstrates how larger samples produce distributions that are more tightly clustered and more closely approximate normality. The image adds minor additional detail by also showing the reduction in spread as n increases. Source.

Between expected successes, expected failures, and the sampling distribution framework, students develop a coherent understanding of why normality checks are essential for valid inference. Ensuring compliance with these criteria gives meaning and reliability to the test statistic and the resulting p-value that underpins statistical decision-making.

Interpreting the Outcome of the Normality Check

Meeting the normality conditions signals that the z-test produces trustworthy results for the given context. Failing the conditions, on the other hand, suggests that the binomial distribution cannot be safely approximated by a normal curve. In such cases, alternative methods should be considered, though these fall outside the scope of this subsubtopic. The primary objective is recognizing that normality verification is not optional but a foundational step in performing valid inference for a population proportion.