Start With the Question’s Target
After identifying the parameter in a mean problem, the next step is to match that target to a procedure. For the decision in this tutorial, ask whether the question concerns one population mean or a difference between two population means. The answer depends on what the question is asking about and how the study’s observations are organized—not simply on how many numbers appear in a data table.
A one-sample t procedure uses one sample to investigate a claim about one population mean, \(\mu\), often by comparing it with a specified benchmark value. A two-sample t procedure compares the means of two independent populations using data from two separate groups. As in “Identifying the Parameter in a Mean Problem,” define the population quantity first; then choose the procedure that addresses it.
“Independent groups” means the observations in one group are not deliberately linked to observations in the other group. For example, one person in a treatment group is not matched with a particular person in a comparison group, and the same person does not contribute one observation to each group. When measurements are linked, the choice requires a separate check; the next tutorial focuses on distinguishing paired data from two independent groups.
A Quick Procedure-Choice Routine
Use this routine before calculating a statistic or entering information in a calculator. It helps prevent the common mistake of choosing a procedure from the number of columns rather than the target parameter and design.
Identify what is measured and its units, as in the earlier tutorial “Identifying the Parameter in a Mean Problem.”
Does it ask whether one population’s mean differs from a fixed value, or whether two populations’ means differ from each other?
A claim about one \(\mu\) points to one-sample t. A comparison of \(\mu_1\) and \(\mu_2\) points to two-sample t if the groups are separate and unlinked.
Determine whether there is one sample, two separate samples, or a design that links observations. If observations are linked, do not automatically call them two independent samples.
The procedure choice identifies the relevant inference method; it does not show that the method’s conditions are satisfied. Check the appropriate conditions before carrying out inference.
A useful clue is the parameter named in the research question. “Is the mean different from 12?” asks about one population mean relative to a fixed value. “Is the mean for Group A different from the mean for Group B?” asks about the difference between two population means. In either case, sample data provide evidence about the target; they are not the target themselves.
One Sample Compared With a Benchmark
A one-sample t procedure is appropriate when the study has one sample from a population and the question concerns that population’s mean. A specified value, such as a standard, target, or claimed mean, provides a comparison point for the hypothesis. It does not create a second sample or a second population mean.
Worked Example: Fill Volume Compared With a Label Claim
A small beverage company wants to know whether its bottles have a mean fill volume different from the stated 500 milliliters. A quality worker randomly selects 12 bottles from one day’s production and measures the fill volume of each bottle. The sample mean is 497.6 milliliters. The 500-milliliter label claim is the comparison value.
Identify the response and population: The quantitative response is fill volume, measured in milliliters. The population of interest is all bottles produced by the company that day. The 12 selected bottles make up one sample from that population.
Identify the target: The question asks whether the population’s mean fill volume differs from 500 milliliters. Let \(\mu\) be the true mean fill volume, in milliliters, for all bottles produced that day. The target is one population mean, \(\mu\), compared with the fixed benchmark 500.
Choose the procedure: This is a one-sample t setting because there is one sample and the target is one population mean. The benchmark is a stated value, not data from a second group of bottles. The sample mean of 497.6 milliliters is a statistic that summarizes the selected bottles; it does not turn the problem into a two-sample comparison.
State what remains to check: The procedure choice alone does not establish whether a one-sample t analysis is appropriate. Before conducting inference, the worker must check the one-sample t conditions for the sampling design and the sample’s distribution. Those checks are a separate step from identifying which mean is being studied.
The same logic applies when a benchmark comes from a regulation, a design target, or an earlier stated standard. Unless the question asks for a comparison between two sampled populations, comparing one sample with a fixed value is not a two-sample t problem.
Two Separate Groups Compared With Each Other
A two-sample t procedure addresses a question about the difference between two population means when the data come from separate, unlinked groups. Define each population mean for the same quantitative response, then keep the group order consistent. As described in “Defining Both Population Means in Context,” the comparison parameter is a difference such as \(\mu_1-\mu_2\).
Worked Example: Two Mulch Types and Seedling Growth
A student growing seedlings wants to compare their mean height after four weeks under two mulch types. The student randomly assigns different seedlings to each mulch type; no seedling receives both types. At four weeks, the sample of seedlings given Mulch A has a mean height of 14.2 centimeters, and the separate sample given Mulch B has a mean height of 12.9 centimeters.
Identify the response and units: The quantitative response is seedling height after four weeks, measured in centimeters. The two groups consist of different seedlings assigned to Mulch A and Mulch B.
Identify the target: Let \(\mu_A\) be the true mean four-week height, in centimeters, for seedlings under Mulch A in the setting of this experiment. Let \(\mu_B\) be the corresponding true mean for seedlings under Mulch B. The research question concerns \(\mu_A-\mu_B\), the difference between two population means.
Check the group structure: Each seedling is in one group only, and no seedling is deliberately matched with a particular seedling in the other group. The groups are separate and unlinked, so this is an independent-groups comparison.
Choose the procedure: A two-sample t procedure is the appropriate choice for investigating the difference between the two population mean heights. The observed difference in sample means is
The 1.3-centimeter result describes the observed samples. The parameter of interest remains \(\mu_A-\mu_B\), the difference in population mean heights. The subtraction order is Mulch A minus Mulch B, so a positive difference indicates a greater mean height for the Mulch A population.
State what remains to check: Choosing two-sample t does not guarantee that its conditions are met. The student must check the design and the data for the conditions associated with a two-sample t procedure, including whether the groups and observations are independent and whether the sample distributions support using the procedure.
One Data Set Can Support Different Questions
The recorded data alone do not always determine the procedure. The research question determines the parameter, and the design determines whether groups are separate or linked. The same collection of measurements might be used to answer different questions, but each question can call for a different parameter and method.
Worked Example: Café Wait Times at Two Times of Day
A café records the wait time, in minutes, for customers served during a morning period and an evening period. The samples contain different customers, with no customers intentionally matched across periods. The morning sample has a mean wait of 6.8 minutes, and the evening sample has a mean wait of 8.1 minutes.
Question 1: What is the mean wait for morning customers? If the target is the true mean wait time for all customers served during the specified morning period, there is one population mean, \(\mu_M\). Using the morning sample to investigate that one mean calls for a one-sample t procedure if the question compares it with a benchmark value, or a one-sample t interval if the goal is to estimate it. The evening sample is not part of this particular target.
Question 2: Do morning and evening customers have different mean wait times? Now the target compares two populations. Let \(\mu_M\) and \(\mu_E\) be the true mean wait times for morning and evening customers in the specified setting. The target is \(\mu_M-\mu_E\), so the relevant comparison is two-sample t because different, unlinked customers contribute to the two samples.
For the comparison question, the observed sample difference is
The negative sign means the morning sample’s mean wait is 1.3 minutes less than the evening sample’s mean wait. It does not establish the size or direction of the population difference with certainty; that is what inference would address. The calculation does not change the procedure choice: it is the two-population comparison question, together with the separate groups, that points to two-sample t.
Full selection: The response is customer wait time in minutes. For Question 1, the target is a single morning population mean, so use a one-sample procedure for that target. For Question 2, the target is a difference between morning and evening population means, and the samples are independent groups, so use a two-sample t procedure. The procedure is selected from the question being answered, not merely from the fact that both columns of data are available.
Benchmarks, Groups, and Linked Measurements
Three situations can look similar because each involves comparing numbers, but they have different targets:
- One sample and a fixed benchmark: The target is one population mean, such as whether the mean fill volume differs from 500 milliliters. This points to one-sample t.
- Two separate, unlinked samples: The target is a difference between two population means, such as Mulch A minus Mulch B. This points to two-sample t.
- Measurements with a deliberate link: If the same units are measured more than once or units are deliberately matched, do not assume the groups are independent. Recognize that the design needs a closer look; the next tutorial examines the distinction between paired t and two-sample t.
Do not decide based only on the number of group labels. A study could record two labels but ask a question about only one group’s mean. Conversely, a study might store data in one column while including a group label for every observation; if the question compares two independent populations, the target is still a difference between two means.
Common Mistakes and AP Exam Tips
- Treating a benchmark as a second group: A stated value such as 500 milliliters is not a second sample or population mean. A full-credit choice explains that the target is one population mean compared with a fixed value.
- Choosing two-sample t whenever two columns appear: The number of columns is not decisive. Identify the parameter in the question and determine which observations contribute to it.
- Choosing one-sample t just because one sample is discussed first: If the question compares that sample’s population with a second population represented by another independent sample, the target is a difference between two means.
- Ignoring how observations were collected: Two sets of measurements are not necessarily independent groups. Check whether the same unit appears in both sets or whether units were deliberately matched. If so, investigate the paired structure rather than automatically choosing two-sample t.
- Defining the statistic instead of the parameter: The difference between sample means, \(\bar{x}_1-\bar{x}_2\), is not the population comparison. For a two-sample question, define \(\mu_1\) and \(\mu_2\), then state the target difference in context.
- Confusing procedure choice with condition checking: Naming one-sample or two-sample t is not a complete inference analysis. After choosing, check the appropriate conditions for the actual sampling or assignment design and the observed data.
For a concise AP response, name the population quantity and connect it to the design: “The question concerns one population mean compared with a fixed value, so a one-sample t procedure is appropriate,” or “The question concerns the difference between two population means from separate, unlinked groups, so a two-sample t procedure is appropriate.” If the observations are paired or matched, say that more design information is needed before treating them as independent groups.
Check Your Understanding
For each situation, identify the target and choose one-sample t or two-sample t when the design supports that choice. Briefly explain your reasoning.
- A random sample of 18 city buses is used to investigate whether the mean fuel efficiency differs from a stated target of 4.5 miles per gallon. Is the target one population mean or a difference between two population means?
- A garden study compares the mean number of flowers on plants receiving two different fertilizers. Different plants receive each fertilizer, with no matching. Which procedure fits the target?
- A school measures the time students spend on a reading task and asks whether the mean for one class differs from a fixed goal of 20 minutes. Does the fixed goal make this a two-sample problem?
- A researcher compares mean weekly exercise time for a sample of adults in one neighborhood with a separate sample from another neighborhood. What information about the design matters when choosing between one-sample and two-sample t?
- A coach records each runner’s time before and after a training plan. Should these observations automatically be treated as two independent groups? What feature of the design should prompt a pause?