Start With the Response, Not the Calculator
A study may report numbers in many forms: individual measurements, averages, counts, or percentages. Those formats alone do not determine the inference procedure. The key question is what response was recorded for each observational unit and what population quantity the research question asks about.
If the response is a quantitative measurement, a question about its population mean generally calls for a t procedure. If the response is categorical with two possible outcomes, such as pass/fail or recovered/not recovered, a question about the population proportion in one category generally calls for a z procedure for proportions. The design then determines whether the procedure is one-sample, paired, or two-sample.
This builds on “Categorical or Quantitative Data: First Decision,” “Identifying the Parameter in a Mean Problem,” and the earlier tutorials on one-sample, paired, and two-sample t procedures. A numerical count can be a quantitative response—for example, the number of customer visits made by each person. But counting how many people said “yes” summarizes a categorical response. Inference depends on the recorded response for each unit, not on whether the final summary contains numbers.
Match the Parameter to the Procedure
A population mean, written \(\mu\), describes the average value of a quantitative variable in a population. A mean procedure uses sample measurements to estimate or test a claim about that average. Because the population standard deviation is usually unknown and estimated with the sample standard deviation \(s\), the procedure uses a t distribution.
A population proportion, written \(p\), describes the fraction of a population in a specified category. For a sample of \(n\) units, the sample proportion is \(\hat{p}=x/n\), where \(x\) is the number of units in that category. A proportion procedure uses the counts in the categories to estimate or test a claim about \(p\), using a z distribution when its conditions are met.
Once the response type and parameter are clear, use the design to select the specific procedure. For quantitative data, one population mean compared with a fixed value calls for a one-sample t procedure; linked measurements call for a paired t procedure on differences; and two independent groups call for an unpooled two-sample t procedure. These choices are developed in “One-Sample t Versus Two-Sample t,” “Paired t Versus Two-Sample t,” and the tutorials on crossover and matched-pairs designs.
For a binary categorical response, one group compared with a fixed proportion calls for a one-proportion z procedure. Two independent groups compared by their proportions call for a two-proportion z procedure. The comparison must concern the proportion in a defined category, not the mean of a numerical measurement.
- One sample, quantitative response, one population mean: one-sample t.
- Two linked quantitative measurements per unit or pair: paired t on the differences.
- Two independent groups, quantitative response, difference in population means: unpooled two-sample t.
- One group, binary categorical response, one population proportion: one-proportion z.
- Two independent groups, binary categorical response, difference in population proportions: two-proportion z.
A request to estimate a parameter points to a confidence interval; a request to assess a claim points to a test. That choice comes after identifying the parameter and design. An interval or test cannot repair a mismatch—for example, a t procedure is not appropriate just because the data were entered as numbers if the response is actually categorical.
Worked Examples: Tell Means and Proportions Apart
Worked Example: Battery Life or Pass Rate?
A quality team selects 12 rechargeable batteries and records each battery’s operating time, in hours. The sample mean is 8.4 hours and the sample standard deviation is 0.9 hour. The team wants to know whether the population mean operating time differs from a stated target of 8 hours. Which procedure matches the question?
Identify the response: The response for each battery is operating time in hours. Time is quantitative, and the question asks about the population mean operating time, \(\mu\).
Match the procedure: There is one sample and its mean is compared with a fixed value. The appropriate family is a one-sample t procedure. The sample standard deviation \(s=0.9\) estimates the unknown population standard deviation; a z procedure for a proportion does not apply.
State the target: If the team is testing the claim of a difference, the hypotheses are \(H_0:\mu=8\) hours and \(H_a:\mu\ne8\) hours. If instead it wants an estimate, it should construct a one-sample t interval for \(\mu\). The response, units, and target remain the same in either case; only the goal changes from testing to estimation.
Now change the research question. Suppose the team records whether each battery meets a specified minimum-life standard: “meets standard” or “does not meet standard.” The response is now categorical, even though operating time may have been used to decide the category. If the question concerns the fraction of all batteries that meet the standard, the parameter is a population proportion \(p\), and a one-proportion z procedure is the appropriate family. The team must use the number meeting the standard out of 12, rather than treating the two labels as measurements for a mean.
Solution: Operating time as a numerical measurement and a target mean call for one-sample t. Meeting or not meeting a standard and a target fraction call for one-proportion z. The wording of the research question and the recorded response—not the shared battery context—separate the procedures.
Worked Example: Testing a Claimed Pass Proportion
A fictional inspection team randomly selects 80 items from a large production run. Each item is classified as pass or fail. Fifty-four pass inspection. The team asks whether the population proportion that passes differs from 0.60. Carry out the appropriate test at \(\alpha=0.05\).
State: Let \(p\) be the true proportion of items in the production run that pass inspection. The hypotheses are \(H_0:p=0.60\) and \(H_a:p\ne0.60\).
Plan: Use a one-proportion z test. The response is categorical with two outcomes, and the question concerns one population proportion compared with a fixed value. The items were randomly selected. Assuming sampling was without replacement, the production run must contain at least 800 items for the 10% condition. Under the null hypothesis, the Large Counts condition is satisfied: \(np_0=80(0.60)=48\) and \(n(1-p_0)=80(0.40)=32\), both at least 10.
Do: The sample proportion is \(\hat{p}=54/80=0.675\). For a one-proportion z test, use the null proportion in the standard error:
For a two-sided alternative, the p-value is approximately \(0.1709\), rounded. Assuming the true pass proportion is 0.60, this is the probability of obtaining a sample proportion at least as far from 0.60 as 0.675, in either direction.
Conclude: Since \(0.1709>0.05\), fail to reject \(H_0\). The data do not provide convincing evidence that the population proportion of items passing inspection differs from 0.60.
This example uses a z procedure because each item contributes a category, pass or fail, and the target is a proportion. If the team instead recorded each item’s precise operating time and asked about average operating time, the question would concern a mean and would call for a t procedure.
Worked Example: Matched Measurements or a Yes/No Outcome?
A school randomly selects 18 students and records each student’s time, in minutes, to complete a practice activity before and after a study-skills workshop. The school asks whether the workshop is associated with a change in mean completion time. A separate question asks whether the proportion of students who finish within 20 minutes changes from before to after.
For completion time: The response is a quantitative measurement in minutes, recorded twice for each student. The two measurements are linked because they come from the same student. Define \(d=\text{time after}-\text{time before}\). The target is the population mean difference \(\mu_d\), so the appropriate procedure is a paired t procedure on the 18 differences, not a two-sample t procedure treating the two sets of times as independent.
For finishing within 20 minutes: Each student’s response at each time is now classified as yes or no. The target concerns proportions, not mean times. However, the before-and-after categories are still linked within students. A two-proportion z procedure is designed for two independent groups, so it is not justified merely because the outcomes are binary. Do not count the before and after responses as if they came from unrelated students. The paired quantitative question clearly uses paired t; the linked categorical question requires attention to a different design and should not be forced into an independent two-proportion procedure.
Solution: The recorded variable and the linkage both matter. Paired numerical times lead to a paired t procedure for \(\mu_d\). Binary outcomes from the same students do not become independent samples just because they can be summarized as two proportions.
Worked Example: Comparing Two Treatments With Two Kinds of Responses
In a fictional randomized experiment, 40 volunteers receive a new treatment and 35 receive a comparison treatment. Researchers record recovery time in hours and whether each volunteer has returned to usual activity by day 5. The sample mean recovery times are 52 hours for the new treatment and 59 hours for the comparison treatment. By day 5, 30 of the 40 volunteers in the new-treatment group and 21 of the 35 in the comparison group have returned to usual activity. Because the participants are volunteers rather than a random sample, conclusions apply to these experimental units; generalizing to a broader population requires an appropriate random-sampling basis.
Question about recovery time: Recovery time is quantitative. The groups contain different volunteers, so the observations are independent rather than paired. The parameter is the difference between the mean recovery times under the two treatments for these volunteer experimental units, \(\mu_1-\mu_2\), where group 1 is the new treatment and group 2 is the comparison treatment. Use an unpooled two-sample t procedure. The observed difference in sample means is \(52-59=-7\) hours, but that statistic alone does not replace the inference procedure.
Question about returning by day 5: The response is categorical: returned or did not return by day 5. The sample proportions are \(30/40=0.75\) and \(21/35=0.60\). The target is \(p_1-p_2\), the difference in proportions returning by day 5 under the two treatments for these volunteer experimental units. Use a two-proportion z procedure, not a two-sample t procedure. For a test of equal proportions, the test’s standard error uses a pooled proportion under the null; for a confidence interval, the standard error is based on the two separate sample proportions.
Solution: The same experiment can require different procedure families for different questions. Mean recovery time calls for unpooled two-sample t. The proportion returning by day 5 calls for two-proportion z. The treatment groups are the same, but the response and parameter differ.
Common Mistakes and AP Exam Tips
- Choosing from the format of a summary: A percentage is often a sample proportion, but a number can also be a quantitative count measured on each unit. Describe the response for one observational unit before choosing.
- Using t because a data set contains numbers: A category might be coded as 0 and 1 for software, but those codes do not turn the response into a quantitative measurement. If the question asks for the fraction in the “1” category, use a proportion procedure.
- Using z for any question with two groups: Two independent groups with quantitative responses and a mean comparison call for two-sample t. Two independent groups with binary categorical responses and a proportion comparison call for two-proportion z.
- Forgetting the design: Two linked measurements are not independent samples. Use paired t for quantitative differences. Do not apply a two-proportion z procedure to paired binary outcomes by pretending the observations are independent.
- Mixing up a mean and a proportion: “Average recovery time” refers to a mean in hours. “Fraction recovered by day 5” refers to a proportion. Name the parameter and its units or category before naming the procedure.
- Choosing the procedure before identifying the goal: First match the response, parameter, and design. Then decide whether the question asks for an estimate (an interval) or evidence about a claim (a test).
For full-credit communication, state the response type, identify the population parameter in context, and name the procedure that matches both the parameter and the design. For example: “Recovery time is quantitative, and the groups are independent, so an unpooled two-sample t procedure will compare the population mean recovery times.” Or: “Returned by day 5 is a binary categorical response, and the groups are independent, so a two-proportion z procedure will compare the population proportions.”
Check Your Understanding
For each situation, identify the response type, parameter, and appropriate procedure family. Note any design feature that affects the choice.
- A random sample of 25 hikers reports the number of kilometers each hiked last weekend. Researchers want to estimate the population mean distance.
- In a random sample of 120 residents, 73 support a proposed park. Researchers test whether the population proportion supporting it is greater than 0.50.
- Two independent groups of plants receive different watering schedules. Researchers compare the mean height in centimeters after six weeks.
- The same 16 students take a timed task before and after a practice session. Researchers ask whether mean completion time changed.
- Two independent groups of customers are asked whether they would recommend a service. Researchers compare the proportions answering yes. Which procedure family fits, and what feature of the design matters?