The Research Question Chooses the Direction
In Writing Null and Alternative Hypotheses for a Proportion, you learned that the null hypothesis gives a reference value for the population proportion, while the alternative describes the departure the test is designed to investigate. This tutorial focuses on choosing that alternative from the wording of the research question.
The choice is between asking whether the population proportion is less than a benchmark, greater than it, or simply different from it in either direction. The first two are one-sided alternatives because they specify one direction. The third is a two-sided alternative because it allows either direction.
To choose, focus on what the researcher wants to find evidence for. Words such as “lower,” “fewer,” or “below” point toward \(p<p_0\). Words such as “higher,” “more,” or “above” point toward \(p>p_0\). Words such as “different,” “changed,” or “not equal” usually point toward \(p\ne p_0\), unless the question explicitly limits the concern to one direction.
The direction belongs to the research question, not to the data. Choose the alternative before examining the sample result. If a sample proportion happens to be above the benchmark, that does not turn a question about possible change in either direction into a greater-than question. The observed result is used later to evaluate the hypotheses, not to decide what question was asked.
A Practical Way to Translate the Wording
First identify the benchmark \(p_0\), then put the research question into plain language without its technical wording. Ask: Is the concern specifically that the proportion is lower? Specifically higher? Or could either an increase or a decrease answer the question? This translation usually makes the symbol clear.
Identify the proportion used as the benchmark, \(p_0\). A claim of 42% corresponds to \(p_0=0.42\).
Ask what result would address the research question: a lower proportion, a higher proportion, or either kind of difference.
Use \(p<p_0\) for a specifically lower proportion, \(p>p_0\) for a specifically higher proportion, or \(p\ne p_0\) for a difference in either direction.
Read the alternative in words. It should say exactly what the research question asks about the population proportion.
A useful final check is to imagine the population proportion is on the opposite side of the benchmark from the one-sided alternative. Would that result answer the research question? If the question specifically asks whether a favorable outcome rate has increased, a decrease does not answer it. If the question asks whether the rate has changed, either an increase or a decrease matters, so a two-sided alternative fits.
Wording That Signals Each Alternative
Some phrases have a fairly direct match to an alternative, but context matters. “Improved” means greater only when improvement has been defined as a higher proportion of successes. If the characteristic is an undesirable outcome, improvement might instead mean a lower proportion. Identify what counts as a success before interpreting the direction.
| Research question wording | Alternative | Meaning in context |
|---|---|---|
| “Is the proportion below 0.42?” | \(H_a:p<0.42\) | The population proportion is less than 0.42. |
| “Has the proportion risen above 0.42?” | \(H_a:p>0.42\) | The population proportion is greater than 0.42. |
| “Has the proportion changed from 0.42?” | \(H_a:p\ne0.42\) | The population proportion could be higher or lower. |
“Changed” generally calls for a two-sided alternative when no direction is named. A change can be an increase or a decrease. By contrast, “has the proportion dropped?” names a direction and calls for a less-than alternative. If a question is vague—for example, “Is the program effective?”—the direction cannot be determined from that wording alone. The researcher needs to specify what outcome defines effectiveness and what kind of change is of interest.
A test with a one-sided alternative does not investigate both sides equally. A less-than alternative is designed to assess evidence of a decrease, not to establish that the proportion is higher. A greater-than alternative assesses an increase, not a decrease. If the research question genuinely treats both directions as important, the alternative should be two-sided.
Worked Examples
Worked Example: Checking Whether a Recycling Rate Is Below a Target
A town’s recycling plan aims for 42% of household waste to be recycled. A committee wants to investigate whether the proportion of households that meet the plan’s recycling target is below 42%. A sample includes 180 households, of which 68 meet the target. Choose the alternative hypothesis. Do not carry out a test.
State: Let \(p\) be the proportion of households in the town that meet the recycling target. The comparison value is \(p_0=0.42\).
Plan: The committee’s question specifically asks whether the proportion is “below 42%.” That wording names a lower direction, so the alternative is one-sided and uses the less-than symbol. The sample count does not determine the direction.
Do: The hypotheses are:
The alternative says that less than 42% of households in the town meet the recycling target. The observed sample proportion is \(68/180\), or about \(0.378\), but that result is not the reason for choosing the less-than alternative. The wording of the committee’s question supplies the direction.
Conclude: A test using this alternative would assess evidence that the town’s population proportion is below 42%. It would not be a test of whether the proportion might instead be above 42%.
Worked Example: Asking Whether a Library Service Has Changed
A library reports that 35% of its cardholders use its digital-book service at least once a month. The library wants to know whether the proportion has changed since the report. In a new random sample of 240 cardholders, 96 report monthly use. Select the alternative and explain why the observed sample result does not make it one-sided.
State: Let \(p\) be the proportion of the library’s cardholders who use its digital-book service at least once a month. The benchmark is \(p_0=0.35\).
Plan: “Has changed” does not specify an increase or a decrease. A proportion above 0.35 and a proportion below 0.35 would both represent a change. Therefore, the question calls for a two-sided alternative.
Do: The hypotheses are:
The sample proportion is \(96/240=0.40\), which is above 0.35. That direction in the sample does not change the alternative. The question asks whether the population proportion has changed in either direction, so the alternative remains \(p\ne0.35\).
Conclude: The two-sided setup investigates whether the proportion of all cardholders who use the service monthly is different from 35%, whether higher or lower. The hypotheses alone do not tell us whether there is convincing evidence of a change.
Worked Example: Testing Whether a Reminder Increases Appointment Attendance
A clinic is considering text reminders for appointments. Without reminders, 78% of scheduled patients attend. The clinic asks whether the proportion attending would be higher with reminders. Choose the alternative. Then explain what kind of result this question is not designed to establish.
State: Let \(p\) be the proportion of patients scheduled at the clinic who attend after receiving a text reminder. The comparison value is \(p_0=0.78\).
Plan: The question asks whether attendance would be “higher.” That is a specific increase, so the alternative is one-sided with a greater-than symbol. The alternative is chosen from the planned research question, before looking at data from the reminder program.
Do: The hypotheses are:
The alternative means that more than 78% of scheduled patients attend after receiving reminders. A result suggesting attendance is lower would not support this alternative, even if it might raise a separate concern worth investigating.
Conclude: This question sets up an investigation of increased attendance, not any possible difference from 78%. If the clinic instead wanted to know whether reminders affect attendance in either direction, the research question and alternative would need to be two-sided.
Worked Example: Interpreting an “At Least” Benchmark
A community garden’s goal is that at least 55% of its members volunteer for a seasonal cleanup. Organizers want to investigate whether the proportion who volunteer falls short of the goal. Choose the alternative hypothesis.
State: Let \(p\) be the proportion of the garden’s members who volunteer for the seasonal cleanup. The goal’s boundary is \(p_0=0.55\).
Plan: The phrase “falls short” means less than the benchmark. For a standard one-proportion test, the null is written at the benchmark equality, and the research question is expressed by the less-than alternative. The everyday wording “at least 55%” does not change the direction being investigated.
Do: The hypotheses are:
The alternative says that fewer than 55% of all garden members volunteer. It does not say that the sample proportion must be below 0.55; it states the population departure the test is designed to assess.
Conclude: The organizers’ question is about whether participation is below the goal, so a less-than alternative matches. If they wanted to know whether the true proportion differs from 55% in either direction, they would need to ask a two-sided question instead.
Common Mistakes and AP Exam Tips
- Letting the sample result choose the tail. A sample proportion above \(p_0\) does not justify switching to \(H_a:p>p_0\) if the original question asked whether the population proportion changed in either direction. Choose the alternative from the research question.
- Treating “different” as “greater.” “Different from 0.42” includes values below 0.42 and above 0.42. Use \(p\ne0.42\), not \(p>0.42\).
- Treating “improved” as automatically greater. Improvement depends on how success is defined. A higher proportion of patients recovering could be an improvement, while a lower proportion experiencing complications could be an improvement. Translate the outcome into the direction of \(p\).
- Using a one-sided alternative for a vague question. If the wording does not say whether an increase or decrease matters, do not guess based on what seems likely. Clarify the research question. If either direction matters, use a two-sided alternative.
- Claiming a one-sided test covers both directions. \(H_a:p<p_0\) investigates a decrease; \(H_a:p>p_0\) investigates an increase. A full-credit explanation states the direction in context and does not claim that the test addresses the opposite direction.
- Confusing the null benchmark with the research direction. In a standard one-proportion test, the null hypothesis is written \(H_0:p=p_0\). The alternative is where the less-than, greater-than, or not-equal direction appears.
Key Takeaway
The alternative hypothesis should express the departure the research question is asking about. A specifically lower proportion calls for a less-than alternative, a specifically higher proportion calls for a greater-than alternative, and a difference in either direction calls for a not-equal alternative.
Check Your Understanding
For each situation, identify the direction requested and write the alternative hypothesis. State whether it is one-sided or two-sided.
- A school wants to know whether the proportion of students who bring a reusable water bottle is greater than last year’s 48%.
- A wildlife team asks whether the proportion of nesting sites occupied by a bird species has changed from 62%, with no direction specified.
- A delivery service wants to investigate whether its late-delivery proportion is below 8% after changing its scheduling system.
- A survey finds a sample proportion above the benchmark, but the research question asks whether the population proportion is different from the benchmark. Should the alternative be changed to greater than? Explain.
- A researcher asks whether a new process “improves quality,” but does not define quality as a higher or lower proportion of successes. What must be clarified before choosing the alternative?