From the Difference Parameter to the Hypotheses
In “The Parameter \(\mu_d\) in Paired Inference,” you defined \(\mu_d\) as the true mean of the paired differences in a population. This tutorial uses that parameter to state hypotheses for a paired t test. The central question is whether the population mean difference is zero or whether it differs from zero in a direction described by the research question.
For a before-and-after study, use the established convention \(d_i=\text{after}_i-\text{before}_i\), unless a problem explicitly specifies another subtraction order. With this convention, a positive difference means a higher measurement after, and a negative difference means a lower measurement after. The hypotheses must use the same difference definition as the data analysis.
The null hypothesis \(H_0:\mu_d=0\) says that the true population mean of the defined differences is zero. For after minus before, this means there is no average change in the population. It does not say that every individual has a difference of zero: some individual differences could be positive and others negative, with a population average of zero.
The alternative hypothesis \(H_a\) states which departure from the null is of interest. Use \(H_a:\mu_d>0\) when the research question specifically predicts a higher population mean after the change. Use \(H_a:\mu_d<0\) when it specifically predicts a lower population mean. Use \(H_a:\mu_d\ne0\) when either a higher or a lower population mean would matter.
Translate the Claim Through the Subtraction Order
Start by stating what a positive difference means. Then translate the research claim into the sign or signs of \(\mu_d\) that would support it. When differences are after minus before, “increased on average” corresponds to a positive mean difference, and “decreased on average” corresponds to a negative mean difference. A claim of “changed” without a predicted direction corresponds to either sign.
The equality in the null hypothesis is important: the null gives a specific reference value against which the sample evidence will be assessed. In the usual before-and-after question about whether there is any average change, that reference is zero. The alternative uses a strict inequality, because it describes values of the parameter that differ from the null in the direction of interest.
- Define \(d_i\) with an explicit subtraction order, such as after minus before.
- Define \(\mu_d\) as the true population mean of those differences, in context and with units.
- Use \(H_0:\mu_d=0\) for a question about no average before-and-after change.
- Select the alternative from the research question, before using the sample results.
- Make sure the sign in \(H_a\) agrees with the stated subtraction order.
As discussed in “One-Sided Versus Two-Sided Alternatives for a Mean,” the choice between one-sided and two-sided alternatives depends on the question, not on what the sample happens to show. If a researcher looks at the sample first and then chooses the direction that makes the result look more persuasive, the test no longer answers the originally planned question in a trustworthy way.
The paired t test treats the list of differences as the quantitative sample. Its hypotheses are about \(\mu_d\), not about the mean before measurement, the mean after measurement considered separately, or the sample mean difference \(\bar d\). The sample statistic \(\bar d\) may help estimate \(\mu_d\), but it is not the parameter named in the hypotheses.
Worked Examples
Worked Example: A Higher Average Reading After Practice
A fictional school program asks whether a short daily practice routine increases students’ average reading speed. Students’ reading speeds, in words per minute, are recorded before and after four weeks of practice for 32 participating students. Define each difference as after minus before. State the parameter and hypotheses.
Solution. The paired unit is a student, because each student contributes a before and an after reading-speed measurement. Let \(\mu_d\) be the true mean after-minus-before difference in reading speed, in words per minute, for all students in the population represented by the study.
With this subtraction order, a positive difference means a student read more words per minute after the practice period than before it. The research question predicts an increase in the population average, so the alternative is higher than zero:
The null says there is no average after-minus-before change in reading speed in the target population. The alternative says the population mean after-minus-before difference is positive, which is the direction that would support an increase on average. The sample size of 32 does not determine the direction; the claim does.
Worked Example: A Lower Average Time After a Schedule Change
A fictional community center changes how participants book appointments. For 18 participants, staff record the time, in minutes, from arrival to the start of an appointment both before and after the change. The question is whether the new schedule reduces the population mean wait time. Use after minus before and state the hypotheses.
Solution. Define \(\mu_d\) as the true mean after-minus-before difference in appointment wait time, in minutes, for all participants in the target population represented by the study. A negative difference means the participant waited fewer minutes after the schedule change than before it.
The question specifically predicts a reduction. Under after minus before, a reduction corresponds to a negative difference, so the alternative is lower than zero:
Here, the null represents no average change in wait time. The alternative represents a negative population mean difference, which corresponds to shorter waits on average after the change. It does not assert that every participant’s wait will be shorter. Some observed differences could be positive even if the population mean difference is negative.
Worked Example: A Change in Either Direction
A fictional greenhouse records the weekly growth, in centimeters, of 25 plants before and after a change in lighting. The staff want to know whether the lighting change affects average weekly growth, but they have no specific prediction about whether growth will rise or fall. Define each difference as after minus before. State the parameter and hypotheses.
Solution. Let \(\mu_d\) be the true mean after-minus-before difference in weekly plant growth, in centimeters, for all plants in the population represented by the study. Positive differences mean greater weekly growth after the lighting change; negative differences mean lower weekly growth.
Because the question asks whether the mean has changed in either direction, both positive and negative values of \(\mu_d\) are relevant. The hypotheses are:
The alternative is two-sided: it includes a population mean difference above zero and one below zero. It is not appropriate to choose \(H_a:\mu_d>0\) merely because a sample happens to show higher growth after the change. The stated question allows either direction, so the hypotheses should allow either direction too.
One-Sided or Two-Sided?
A one-sided alternative is appropriate when the research question identifies one direction as the relevant evidence. For example, if shorter wait times are the specific goal, define after minus before and use \(H_a:\mu_d<0\). If higher reading speed is the specific goal, the same subtraction convention leads to \(H_a:\mu_d>0\).
A two-sided alternative is appropriate when a difference in either direction would answer the question. A process change might be undesirable whether it raises or lowers an important measurement, or researchers might simply ask whether the population average changed without predicting the direction. In those cases, use \(H_a:\mu_d\ne0\).
A one-sided alternative is not a general shortcut for asking whether “something happened.” It makes a directional claim. If the observed sample mean difference points in the direction opposite to that claim, the one-sided test does not switch direction after the fact. This is why the direction should be set using the research question before examining the results.
Common Mistakes and AP Exam Tips
- Using the wrong parameter. The hypotheses concern \(\mu_d\), the true population mean of the paired differences. They do not concern \(\bar d\), which is the sample mean, or a mean from just the before or after column.
- Reversing the sign. With after minus before, a decrease points to a negative mean difference. Check the subtraction order before writing the alternative.
- Writing an inequality in the null. For this test, write \(H_0:\mu_d=0\). Put the directional or “not equal” statement in \(H_a\).
- Using a two-sided alternative for a specific directional claim. If the question is specifically whether a measurement increased, use the positive direction under after minus before. If either direction matters, use the two-sided alternative.
- Choosing the direction after seeing the data. The research question determines \(H_a\), not whether the observed \(\bar d\) happens to be positive or negative.
- Claiming every pair must change in the predicted direction. The alternative is about the population mean difference. Individual paired differences can vary in sign.
- Leaving the setting unclear. Define \(\mu_d\) with the measurement, units, population, and subtraction order so the symbols have a clear interpretation.
For a full-credit response, identify the population mean difference in context, state the subtraction order, write the null equality and the alternative that matches the question, and briefly connect the sign to an increase, decrease, or change in either direction. The conditions for using paired inference are checked separately, as in “Conditions for One-Sample Versus Paired Data”; they do not determine which alternative hypothesis the research question calls for.
Check Your Understanding
For each scenario, use after minus before unless another subtraction order is stated.
- A study asks whether a new stretching routine increases average flexibility. Write the alternative hypothesis and explain the sign.
- A clinic asks whether a reminder system reduces the average number of minutes patients wait. Write the null and alternative hypotheses.
- A researcher asks whether a change in classroom lighting affects average task-completion time, without predicting a direction. Which alternative is appropriate?
- For a paired study, define \(d_i=\text{before}_i-\text{after}_i\). If the question is whether the after measurement is higher on average, should \(\mu_d\) be greater than or less than zero under this order?
- Why should a student not change from a two-sided alternative to a one-sided alternative after seeing that \(\bar d\) is positive?