Writing Null and Alternative Hypotheses for a Proportion

From a Claim to Two Hypotheses

In The Logic of a Significance Test, you learned that a significance test begins by treating a null claim as true and asking whether the observed data would be unusual under that claim. Before a test can do that, its hypotheses must clearly identify the population proportion being studied and the value or direction being assessed.

For a one-proportion test, the parameter is \(p\), the true proportion of a specified population with a specified characteristic. The sample proportion, \(\hat{p}\), is calculated from observed data. Hypotheses are statements about the population parameter \(p\), not about the sample proportion and not about a particular sample count.

Definition: The null hypothesis, \(H_0\), gives the reference value for the population proportion. For a standard one-proportion test, write it as \(H_0:p=p_0\), where \(p_0\) is the hypothesized proportion. The alternative hypothesis, \(H_a\), states the direction or kind of departure from that value that the test is designed to investigate.

The equality belongs in the null hypothesis. The alternative uses one of three inequality symbols: \(>\), \(<\), or \(\ne\). In words, these mean “greater than,” “less than,” and “different from,” respectively. Which one fits depends on the question being asked; the next tutorial focuses on choosing between these alternatives.

H_0:p=p_0 \qquad H_a:p>p_0,\quad H_a:p<p_0,\quad\text{or}\quad H_a:p\ne p_0

A useful habit is to write the meaning of \(p\) in a sentence before writing either hypothesis. Name the population and the characteristic that counts as a success. For example, if the question concerns customers who prefer a new package design, define \(p\) as the proportion of customers in the target market who prefer that design. This definition keeps the hypotheses connected to the real question.

The value in the null hypothesis is a population claim. If a company claims that 60% of its customers prefer a new package, the null value is \(p_0=0.60\), so the null hypothesis is \(H_0:p=0.60\). Writing \(p=60\) would be incorrect: proportions are written on a scale from 0 to 1. In words, \(p=0.60\) means that 60% of the specified population has the characteristic.

Writing Each Part Precisely

A complete hypothesis setup has three connected pieces: a definition of the parameter, the hypotheses in symbols, and a statement of the hypotheses in context. The symbolic form makes the mathematical claim unambiguous. The words show that you understand what population and characteristic the claim refers to.

Define \(p\).
Name the population and the characteristic of interest. Make clear what counts as a success.

Identify \(p_0\).
Find the numerical proportion in the reference claim, and express a percentage as a proportion when writing the symbols.

Write the null hypothesis.
Use \(H_0:p=p_0\). In words, state that the population proportion equals the claimed value.

Match the alternative to the question.
Use \(>\), \(<\), or \(\ne\) as appropriate, then describe that relationship in words and in the context.

The alternative is not simply “the claim is false.” It gives a more specific description of the evidence the test will look for. If the question asks whether a proportion is higher than the reference value, the alternative says it is greater. If the question asks whether it has changed in either direction, the alternative says it is different. The direction is part of the question, not something chosen after seeing which way the sample result went.

Keep sample information out of the hypotheses. Suppose a random sample contains 124 customers who prefer the new design. That result may later be used to calculate \(\hat{p}\) and assess the hypotheses, but it does not change the hypotheses themselves. They describe the population claim and the research question, and should be set before using the observed result to judge evidence.

Key distinction: Hypotheses describe a population parameter. Sample results describe the data collected. Write \(H_0\) and \(H_a\) using \(p\) and the claim—not \(\hat{p}\), the sample count, or a value selected to match the sample.

Worked Examples

Worked Example: Testing a 60% Customer-Preference Claim

A company says that 60% of customers in its target market prefer its redesigned drink bottle to the current design. The company wants to investigate whether the true proportion who prefer the redesigned bottle is greater than 60%. A random sample of 200 customers will be surveyed. Write the hypotheses in symbols and words. Do not carry out the test.

State: Let \(p\) be the proportion of all customers in the company’s target market who prefer the redesigned bottle to the current design. This defines the population and the characteristic being counted.

Plan: The company’s reference claim is 60%, which is \(0.60\) as a proportion. The question asks whether the population proportion is greater than that value. The sample size does not determine either hypothesis.

Do: The null hypothesis uses equality at the claimed value, and the alternative uses “greater than” to match the question:

H_0:p=0.60 \qquad H_a:p>0.60

In words, the null hypothesis is that 60% of all customers in the target market prefer the redesigned bottle. The alternative hypothesis is that more than 60% of all customers in the target market prefer it.

Conclude: These hypotheses set up a test of the company’s 60% reference value against the question of whether the population proportion is higher. No conclusion about the company’s claim can be made from the hypotheses alone; the sample data and an appropriate test would be needed to assess evidence.

Worked Example: Asking Whether the Preference Proportion Has Changed

The same company wants to know whether the proportion of target-market customers who prefer its redesigned bottle is different from 60%—either higher or lower. A survey finds that 124 of 200 sampled customers prefer the redesigned bottle. Write the hypotheses and explain whether the sample result changes them.

State: Let \(p\) be the proportion of all customers in the company’s target market who prefer the redesigned bottle.

Plan: The stated reference value is \(p_0=0.60\). The question asks about a difference in either direction, so the alternative should use \(\ne\). The observed count, 124 out of 200, is sample information; it does not decide which alternative matches the question.

Do: In symbols, the hypotheses are:

H_0:p=0.60 \qquad H_a:p\ne0.60

In words, the null hypothesis is that 60% of all customers in the target market prefer the redesigned bottle. The alternative hypothesis is that the proportion of all customers in the target market who prefer it is different from 60%.

Conclude: The hypotheses are the same whether the sample contains 124, 100, or 150 customers who prefer the bottle: the research question and the claimed benchmark determine their form. The sample result will be used later to evaluate the hypotheses, not to rewrite them. This setup alone does not establish whether the proportion is different from 60%.

Worked Example: Checking a Claim That at Least 60% Prefer a Service

A transit company advertises that at least 60% of riders prefer receiving service alerts by text rather than by email. A customer research team wants to check whether the proportion who prefer text alerts is below 60%. In a random sample of 150 riders, 76 prefer text alerts. Write the hypotheses in symbols and words.

State: Let \(p\) be the proportion of all riders served by the company who prefer receiving service alerts by text rather than by email.

Plan: “At least 60%” is the company’s wording, but the research question asks whether the proportion is below 60%. For a standard one-proportion test, use the boundary value \(0.60\) in the null hypothesis and express the research question with a less-than alternative. The sample count of 76 does not determine the direction.

Do: The hypotheses are:

H_0:p=0.60 \qquad H_a:p<0.60

In words, the null hypothesis is that 60% of all riders served by the company prefer text alerts. The alternative hypothesis is that less than 60% of all riders served by the company prefer text alerts.

Conclude: This setup investigates whether the population proportion is below the 60% benchmark. It does not say that the sample proportion is below 60%, nor does it show that the company’s advertising is correct or incorrect. Those judgments require evaluating the sample evidence with an appropriate procedure and its conditions, as discussed in the earlier tutorials on one-proportion inference.

Common Mistakes and What Full Credit Requires

Putting the sample proportion in the hypotheses. A hypothesis concerns the population proportion \(p\). The sample proportion \(\hat{p}\) is evidence used in the test, not the parameter being tested.
Leaving equality out of the null hypothesis. For a standard one-proportion test, write \(H_0:p=p_0\). Do not write the null as \(p\ne p_0\), \(p>p_0\), or \(p<p_0\).
Using the wrong scale for a percentage. A 60% claim is \(0.60\) in the symbolic hypotheses, not 60. In words, you may express the same value as 60%.
Forgetting to define the population. “\(p\) is the proportion who prefer the new bottle” is incomplete if it does not identify whose preferences are being studied. Name the target population.
Writing vague hypotheses in words. “The proportion is different” does not say which proportion or what it differs from. A full-credit statement identifies the population, characteristic, and benchmark.
Choosing the alternative to fit the sample result. The alternative comes from the research question and should be set before using the data to decide what the evidence shows. A sample proportion above 60% does not, by itself, justify changing a two-sided question into a “greater than” question.
Claiming the hypotheses prove a conclusion. Writing hypotheses only defines what a test will assess. It does not establish that the null is true, that the alternative is true, or that the data provide convincing evidence.

AP Exam Tip: For full-credit communication, define \(p\) in context, write \(H_0:p=p_0\), write the alternative that matches the stated question, and translate both hypotheses into complete sentences. Use the population—not the sample—as the subject of those sentences.

Key Takeaway

A well-written hypothesis setup begins by defining the population proportion and identifying the benchmark in the claim. The null hypothesis uses equality at that benchmark; the alternative states the departure that the question asks the test to investigate.

Key takeaway: Define \(p\) in context, write \(H_0:p=p_0\), and choose \(H_a\) to match the research question. Hypotheses describe the population; sample results provide evidence for evaluating them.

Check Your Understanding

For each item, define the population proportion and write both hypotheses in symbols and in words.

A bakery claims that 60% of its customers prefer its new bread recipe. Researchers want to know whether the true proportion is greater than 60%.
A park department wants to know whether the proportion of local visitors who use a trail has changed from 35%.
A phone company claims that at least 70% of its customers are satisfied with its support service. Researchers want to investigate whether the proportion is below 70%.
Explain why a sample proportion of 0.64 should not be used in place of the 0.60 claim when writing the null hypothesis.
In a sentence, distinguish the population proportion \(p\) from the sample proportion \(\hat{p}\).