This subsubtopic develops a precise understanding of how to compute a test statistic for a population proportion and how the resulting p-value measures the extremeness of sample results.

Calculating Test Statistics and p-Values

A significance test for a population proportion begins by quantifying how far the observed sample proportion lies from the hypothesized value under the null model. This distance, standardized by expected variability, forms the basis of the test statistic, which follows a known distribution when the null hypothesis is assumed to be true.

The Role of the Test Statistic

The test statistic measures how inconsistent the sample evidence is with the null hypothesis value of the population proportion. It compares the difference between the observed proportion and the null hypothesized proportion in standard deviation units.

The test statistic uses the properties of the standard normal distribution when appropriate conditions are met. This connection allows researchers to use tail probabilities from the normal distribution to compute p-values.

Computing the z-Statistic for a Population Proportion

The standardized form of the test statistic uses the hypothesized proportion to determine expected sampling variability.

EQUATION

Because the formula uses $p_{0}$ rather than $\hat{p}$ in the standard error, the z-statistic reflects sampling variation expected if the null hypothesis were true. This assumption ties directly to the logic of inference. This formula measures how many standard deviations the observed sample proportion is from the hypothesized population proportion under the null hypothesis.

This figure displays the test statistic for a one-sample z-test for a population proportion, showing each component of the formula clearly. It reinforces the idea that the z-statistic standardizes the difference between the sample proportion and the hypothesized proportion using the standard deviation of the sampling distribution. The color styling is decorative and not required by the syllabus but does not alter the statistical meaning. Source.

Understanding Sampling Variability Under the Null

The behavior of the test statistic is interpreted relative to the null distribution, which describes the distribution of the test statistic assuming the null hypothesis is accurate. The z-statistic should follow a standard normal distribution when conditions justify the use of normal approximation. This approximation allows researchers to determine how “extreme” the observed result is in comparison with what is expected under the null hypothesis.

This distribution provides the basis for converting the test statistic into a probability statement.

Determining Extremeness Through p-Values

The p-value quantifies how surprising the observed test statistic would be if the null hypothesis accurately described the population. It represents a probability, not a measure of effect size or practical importance. Instead, it reflects the compatibility of the sample data with the null hypothesis.

Geometrically, the p-value is the area in the tail or tails of the standard normal curve beyond the calculated z-statistic, under the assumption that the null hypothesis is true.

This diagram represents the standard normal distribution used as the null distribution in a significance test. The dashed line marks the observed test statistic, and the shaded region represents the p-value for a right-tailed test. Although stylistically simplified, it illustrates exactly the same tail-area interpretation used for p-values in one-proportion z-tests. Source.

A test with a smaller p-value indicates stronger evidence against the null hypothesis because such an extreme result is less likely under the null model. A larger p-value suggests the observed sample outcome is not unusual given the hypothesized parameter.

Direction of the Alternative Hypothesis and Tail Areas

The method for computing the p-value depends on the form of the alternative hypothesis. Understanding which tail—or tails—of the distribution represent more extreme outcomes is essential.

• For greater-than alternatives (H_{a}: p > p_{0}):

  • The p-value corresponds to the upper tail area beyond the observed z-statistic.

• For less-than alternatives (H_{a}: p < p_{0}):

  • The p-value corresponds to the lower tail area beyond the observed z-statistic.

• For not-equal-to alternatives ($H_{a}: p \neq p_{0}$):

  • The p-value is the sum of both tail areas, capturing extremeness in either direction.

These tail interpretations ensure alignment between the research question and the probability being assessed.

Connecting Test Statistics and Inference

Together, the test statistic and p-value provide a framework for evaluating the plausibility of the null model. The z-statistic offers a standardized numerical summary of sample evidence, while the p-value translates that evidence into a probabilistic measure. Both components rely on assuming the null hypothesis is true—a fundamental aspect of significance testing that supports valid inference about population proportions.