Hypothesis formulation establishes the foundation of statistical testing by defining the claims being evaluated about a population proportion and determining the direction of the proposed alternative.

The Purpose of Hypothesis Formulation

Formulating hypotheses creates a structured comparison between an assumed population proportion and the value suggested by sample evidence. In inference for a single proportion, this framing determines how evidence will be evaluated using a one-sample z-test for a population proportion.

Before constructing hypotheses, it is essential to understand that every significance test begins with a claim treated as “innocent until proven guilty.” This role belongs to the null hypothesis, which contrasts with the alternative hypothesis, the claim for which the researcher seeks supporting evidence.

The Null Hypothesis

The null hypothesis (H0) represents the status quo, a benchmark proportion, or a statement of no change. It assigns a specific numeric value to the population proportion.

The null is expressed using an equality: H0: p = p0. It does not reflect what the researcher hopes to prove; instead, it provides a reference against which evidence is tested. Selecting $p_0$ requires contextual relevance, such as a historical rate or a regulatory standard.

Hypothesis formulation uses the null as the anchor for calculating expected outcomes and assessing how unusual the sample proportion would be if the null were correct.

The Alternative Hypothesis

The alternative hypothesis (Ha) makes a directional or non-directional claim about the population proportion and identifies what the researcher aims to support with data. It must contrast meaningfully with the null.

The alternative is always written with a strict inequality and chooses one of three forms:

One-Sided (Greater Than or Less Than)

A one-sided alternative states that the population proportion is either greater than or less than the hypothesized value. It is used when context implies the difference can only logically occur in one direction.

Examples include:

• Ha: p > p0

  Indicates interest in whether the proportion exceeds the hypothesized rate.

• Ha: p < p0 Indicates interest in whether the proportion falls below the hypothesized rate.

These forms limit the “extreme outcomes” considered when calculating the p-value to one tail of the sampling distribution.

This diagram compares left-tailed, right-tailed, and two-tailed hypothesis tests by shading rejection regions where H0H_0H0​ would be rejected. It visually connects each form of HaH_aHa​ to the part of the sampling distribution that provides evidence against the null. Some labels illustrate concepts that appear in later inference topics beyond this specific subsubtopic. Source.

Two-Sided (Not Equal To)

A two-sided alternative states that the population proportion is simply different from the hypothesized value, without committing to a direction.

• Ha: p ≠ p0

This form is appropriate when any deviation from the null value is meaningful. Two-sided tests are more conservative because evidence must be strong enough to support a departure on either side.

This diagram presents a two-tailed hypothesis test with symmetric shaded rejection regions representing where results would be considered too extreme under H0H_0H0​. It shows that evidence against the null can occur in either direction. The figure also includes critical value markers relevant to inference topics that extend beyond this subsubtopic. Source.

How Hypotheses Work Together

The hypotheses always appear as a pair, and together they establish the assumptions needed for inference:

• H0 specifies what is assumed to be true.

• Ha specifies what the researcher seeks to show.

Because all calculations for the test statistic and p-value are based on the assumption that H0 is true, the null hypothesis affects the structure of the sampling distribution used for inference.

Establishing clear hypotheses ensures the test is logically aligned with the research question. A mismatch between the question and the form of Ha can lead to incorrect interpretation or loss of statistical power.

Choosing the Correct Form of the Alternative

Selecting an appropriate alternative requires attention to study context:

• Use Ha: p > p0 when increases are meaningful or expected.

• Use Ha: p < p0 when decreases are meaningful or expected.

• Use Ha: p ≠ p0 when any difference matters, or when direction is not predictable beforehand.

Researchers should determine the direction before collecting data to avoid bias. Changing Ha after seeing the data undermines the validity of the test.

Role of Hypothesis Formulation in Inference

Hypothesis formulation does not determine the outcome of the test but instead shapes how the evidence is evaluated. Once H0 and Ha are established:

• The null defines the expected sampling distribution.

• The alternative determines where evidence is sought (upper tail, lower tail, or both tails).

• The significance test later compares observed results to these expectations.

Because the entire inferential process depends on correctly written hypotheses, this step is one of the most critical in performing a test for a population proportion.

This clear structure—H0 as an equality and Ha as a strict inequality—ensures that statistical reasoning stays consistent with the AP Statistics framework and supports reliable decision-making in context.