Start by Defining What \(p\) Describes
In Writing Null and Alternative Hypotheses for a Proportion and Choosing One-Sided or Two-Sided Alternatives, you learned to write hypotheses about a population proportion and choose an alternative that matches the research question. Before writing those symbols, you need to define exactly which population proportion they refer to.
A hypothesis such as \(H_0:p=0.60\) is incomplete unless the reader knows what \(p\) measures. The symbol should stand for one specific proportion in one specified population, with a clearly stated characteristic. A precise definition helps ensure that the hypotheses answer the intended question rather than a related but different one.
A reliable form is: “Let \(p\) be the true proportion of all [specified population] who [have the characteristic].” For example, “Let \(p\) be the true proportion of all customers who use the store’s mobile app at least once a week.” The statement tells us who is counted and what counts as having the characteristic.
The word true signals that \(p\) describes the population, not just the people observed in a sample. The value of \(p\) is generally unknown, which is why the study collects data. The sample proportion, \(\hat{p}\), summarizes the sample data; it is not the parameter in the hypotheses. As emphasized in Writing Null and Alternative Hypotheses for a Proportion, hypotheses describe the population, while sample results provide evidence about those hypotheses.
Include the Population and the Characteristic
A good parameter definition answers two questions. First, which individuals or items are in the population? Second, what characteristic makes an individual or item count? Leaving out either part can make \(p\) ambiguous.
The population should match the question the study is intended to answer. It might be all current customers of a business, all registered voters in a town, or all appointments scheduled at a particular clinic during a specified period. If the conclusion is meant to concern only a particular group, name that group rather than a broader population.
The characteristic should be specific enough that each member of the population either has it or does not. “Likes the service” may be too vague unless the study explains how liking is measured. “Gives the service a rating of at least 4 on a 5-point survey question” is more precise. For a one-proportion setting, that definition separates observations into successes and failures, as in the earlier tutorial Counting Successes and Failures Correctly.
When useful, include a location, time period, or eligibility rule. “All subscribers” might mean current subscribers on a particular date, people who subscribed at any time during a year, or subscribers to one service plan. Naming the relevant group prevents different readers from interpreting \(p\) differently.
The definition should not include the number sampled or describe only the sample. If 48 of 120 surveyed customers use an app weekly, \(48/120\) is the observed sample proportion. It may be used to calculate \(\hat{p}\), but \(p\) remains the proportion among the specified population of customers. Defining \(p\) as “the proportion of the 120 surveyed customers who use the app” would change the target from the population parameter to a feature of the data already collected.
Be Careful About “Among” and Subgroups
Some questions ask about a proportion within a particular subgroup. In that case, state the subgroup as the population for the parameter. The phrase among all members of that subgroup makes clear which individuals belong in the denominator.
For instance, a question about the percentage of patients who attend among patients who received a reminder is not asking for the percentage of all scheduled patients who both received a reminder and attended. The first proportion uses reminded patients as its population and attendance as its characteristic. The second uses all scheduled patients as its population and the combination of receiving a reminder and attending as its characteristic. Those are different parameters and answer different questions.
To check the denominator, ask: “Out of which group are we finding a proportion?” Then ask: “What counts as a success within that group?” These questions are particularly useful when eligibility, treatment, participation, or exposure determines which people the question concerns.
A Short Routine for Defining \(p\)
Name the full group the research question is about. If the question is about a subgroup, specify that subgroup as the population for this proportion.
State exactly what an individual must do, have, or report to count as having the characteristic.
Use “Let \(p\) be the true proportion of [population] who [characteristic].” Add a time period or other boundary when it matters.
Use \(p\), not \(\hat{p}\), in the hypotheses. Check that the null benchmark and alternative describe the defined population proportion.
The last step matters because symbols do not repair a mismatched definition. If the question asks whether a subgroup’s attendance rate exceeds a benchmark, the definition and hypotheses must both concern attendance within that subgroup. A correctly chosen greater-than sign attached to the wrong population still gives the wrong setup.
Worked Examples
Worked Example: Defining a Customer Proportion
A grocery chain wants to test whether more than 60% of its current customers use its self-checkout service at least once a month. A random sample of customers will be surveyed. Define the parameter and write the hypotheses.
Define: The population is all current customers of the grocery chain. The characteristic is using the self-checkout service at least once a month.
Match: The question asks whether the population proportion is greater than 0.60. The sample will provide evidence, but it does not define the parameter.
Write: Let \(p\) be the true proportion of all current customers of the grocery chain who use its self-checkout service at least once a month. Then:
Check: Both hypotheses concern the proportion among all current customers, and the characteristic is the same in the definition and the research question. The parameter is \(p\), not the proportion found in the sample.
Worked Example: Defining Attendance Among Patients Who Receive Reminders
A clinic wants to know whether more than 82% of patients who receive a text reminder attend their scheduled appointment. Define the parameter and write the hypotheses. Be precise about which patients are in the population.
Define: The population is all patients scheduled for an appointment at this clinic during the study period who receive a text reminder. The characteristic is attending the scheduled appointment.
Match: The phrase “patients who receive a text reminder” identifies the group among whom the attendance proportion is calculated. It is not asking for the proportion of all scheduled patients who both receive a reminder and attend.
Write: Let \(p\) be the true proportion of patients scheduled at this clinic during the study period who receive a text reminder and attend their appointment, among all scheduled patients who receive a text reminder. The benchmark is 0.82, and the question asks whether the proportion is higher:
Check: The denominator is the group of scheduled patients who receive reminders; attendance is the characteristic being measured within that group. If instead \(p\) were defined as the proportion of all scheduled patients who both received a reminder and attended, the hypotheses would address a joint proportion and would not answer the clinic’s stated question.
Worked Example: Naming a Population and a Precise Outcome
A regional transit agency is investigating whether the proportion of weekday riders who arrive at their stop within five minutes of the posted time differs from 70%. Define the parameter and write the hypotheses.
Define: The population is all weekday riders using the agency’s buses during the month being studied. The characteristic is arriving at the rider’s stop within five minutes of the posted time. This gives a specific criterion for counting a rider as a success.
Match: “Differs from 70%” allows the true proportion to be either below or above 0.70. As in Choosing One-Sided or Two-Sided Alternatives, that wording calls for a two-sided alternative.
Write: Let \(p\) be the true proportion of all weekday riders using the agency’s buses during the study month who arrive at their stop within five minutes of the posted time. The hypotheses are:
Check: The definition identifies the riders, the time boundary, and the success criterion. It does not define \(p\) as the proportion of riders in the sample who meet the criterion; that observed proportion is \(\hat{p}\).
Worked Example: Revising a Vague Parameter Definition
A community center plans a test of whether fewer than 40% of its members attend a weekend workshop at least once during the winter season. A student writes, “Let \(p\) be the proportion who attend.” Revise the definition and write the hypotheses.
Identify what is missing: “The proportion who attend” does not specify which people are being counted, what kind of attendance qualifies, or the relevant time period.
Define: The population is all members of the community center during the winter season. The characteristic is attending at least one weekend workshop during that season.
Write: Let \(p\) be the true proportion of all community center members during the winter season who attend at least one weekend workshop during that season. Since “fewer than 40%” names a lower direction, the hypotheses are:
Check: The hypotheses refer to the proportion of all members in the stated population who meet the stated attendance criterion. A sample of members may be used to assess the claim, but the sample itself is not the population described by \(p\).
Common Mistakes and What Full Credit Says
- Defining \(p\) with \(\hat{p}\). Writing “Let \(p\) be the proportion in the sample who…” defines a sample result, not the population parameter. A full-credit definition says “the true proportion of [specified population] who [characteristic].”
- Leaving the population unstated. “Let \(p\) be the proportion who use the service” does not say whose use is being measured. Name the customers, riders, members, or other target group.
- Leaving the characteristic vague. “Satisfied,” “successful,” or “attends regularly” may need a measurable definition. State the criterion that counts as having the characteristic.
- Using the wrong denominator for a subgroup question. If the question asks for an outcome proportion among people who received a treatment or reminder, name that subgroup as the population. Do not silently replace the question with the proportion of everyone who both received the treatment and had the outcome.
- Using a sample count to define the parameter. A statement such as “the proportion of the 90 people surveyed who responded yes” describes the observed sample. The parameter concerns the specified population from which the sample was drawn.
- Changing the definition between the parameter and the hypotheses. If the definition is about customers who use a service monthly, the hypotheses must test that same monthly-use proportion—not weekly use or use at any time.
Key Takeaway
A hypothesis test for one proportion concerns a specific, fixed population proportion. Defining \(p\) precisely makes clear whose proportion is being tested, what counts as a success, and whether the research question is about a whole population or a subgroup.
Check Your Understanding
For each situation, write a contextual definition of \(p\). Where requested, also write the hypotheses.
- A school asks whether more than 52% of its current tenth-grade students have a part-time job. State a suitable definition of \(p\) and the hypotheses.
- A parks department asks whether the proportion of visitors who recycle a bottle is below 30%, among visitors who buy a drink at a park kiosk. What is the population for \(p\), and what is its characteristic?
- A student defines \(p\) as “the proportion of the 75 survey respondents who prefer evening classes.” What is wrong with this definition for a population hypothesis test? Rewrite it for a question about all currently enrolled adult learners at the school.
- A clinic asks whether the proportion of patients who attend an appointment differs from 90%, among patients who receive a reminder. Explain why defining \(p\) as the proportion of all scheduled patients who both receive a reminder and attend would not match the question.
- A research question concerns whether the proportion of all neighborhood households that compost food scraps has decreased from 45%. Write the contextual definition and both hypotheses. Do not use a sample proportion in the definition.