Choose the Procedure Before You Calculate
A mixed set of mean-inference questions asks you to do more than recognize a procedure name. You must connect the research question to a population parameter, distinguish the study designs, decide whether the goal is a test or an interval, and check the conditions for the method that fits. The selection flowchart in “A Decision Flowchart for Mean Inference” and the writing guidance in “Selecting and Justifying a Procedure in Writing” provide a foundation for this practice.
For all three common mean procedures in this tutorial, the response variable is quantitative and the analysis uses a t distribution. What changes is the target and the way observations are related. A fixed benchmark gives one-sample inference. Genuine links between two measurements give paired inference on differences. Two unlinked groups give independent two-sample inference. The question’s goal then determines whether to use a test or an interval.
Keep the order of a difference consistent throughout. For a paired procedure, define \(d\) in words before using \(\mu_d\). For two independent groups, state which population is group 1 and which is group 2 before writing \(\mu_1-\mu_2\). Reversing the order changes the sign of the parameter, statistic, and interval endpoints.
A Quick Comparison of the Three Setups
| Question structure | Population target | Data analyzed | Procedure family |
|---|---|---|---|
| One sample compared with a fixed value | \(\mu\) | Individual measurements | One-sample t test or interval |
| Two measurements linked by unit or match | \(\mu_d\) | One difference per pair | Paired t test or interval |
| Two unlinked groups | \(\mu_1-\mu_2\) | Each group separately | Unpooled two-sample t test or interval |
This table is a selection aid, not a substitute for the condition checks. As in “Matching Procedures to Conditions,” the shape evidence must match what the procedure analyzes: individual values for one-sample t, differences for paired t, and values in each group for two-sample t. If the prompt supplies no relevant plot or summary, do not claim that the data shape satisfies a condition.
Worked Example: One Sample Versus a Fixed Target
A fictional food-testing lab randomly selects 25 sealed juice bottles from a production lot of 500. The technicians measure the amount of juice in each bottle, in milliliters. The sample mean is 498.6 mL and the sample standard deviation is 3.5 mL. A plot of the measurements is roughly symmetric with no apparent outliers. The lab asks whether the population mean fill is below the 500 mL target. Use \(\alpha=0.05\).
State: Let \(\mu\) be the true mean fill, in milliliters, of bottles in this production lot. Since the question asks whether the mean is below the target, the hypotheses are \(H_0:\mu=500\) and \(H_a:\mu<500\).
Plan: Use a one-sample t test for \(\mu\). There is one sample of quantitative measurements compared with a fixed target; 500 mL is not a second sample. The bottles were randomly selected. The 10% condition is satisfied because \(25\leq0.10(500)=50\). The roughly symmetric distribution with no apparent outliers supports using a one-sample t procedure.
Do: The standard error and test statistic are
The degrees of freedom are \(25-1=24\). For the left-tailed alternative, the p-value is approximately \(0.0285\), rounded. Since \(0.0285<0.05\), reject \(H_0\).
Conclude: The data provide convincing evidence that the mean fill of bottles in this production lot is below 500 mL.
The direction in the alternative matters: a negative \(t\) statistic supports this left-tailed question, and the p-value counts results at least that far below the null value. The procedure does not test whether each bottle contains less than 500 mL; its parameter is the population mean.
Paired Data: Make the Difference the Sample
When observations are paired, first state the order of the difference, then treat the collection of differences as one sample. Do not choose the paired method merely because two columns appear in a table. The design must link the observations, as emphasized in “Spotting Paired Designs in Word Problems.” Check the shape of the differences, not the two original columns separately.
Worked Example: A Paired t Interval for Mean Change
A fictional sports-science class randomly selects 10 runners from 120 members of a running club. Each runner completes a short course using two shoe designs, with the order randomized. Let \(d=\text{time in design A}-\text{time in design B}\), in seconds. Positive values mean the runner was faster in design B. The sample mean difference is 1.8 seconds and the sample standard deviation of the differences is 2.4 seconds. The differences are roughly symmetric with no apparent outliers. The class wants a 95% confidence interval for the population mean difference.
Identify: The same runners used both designs, so the measurements are paired. The target is \(\mu_d\), the true mean difference in course time for the population represented by the random sample, using the stated order. Because the goal is to estimate that difference rather than test a claim, use a paired t interval.
Check conditions: The runners were randomly selected, and the 10% condition holds because \(10\leq0.10(120)=12\). Each runner contributes one difference, and the differences are roughly symmetric with no apparent outliers. These facts support paired t inference.
Calculate: With \(n=10\), the degrees of freedom are \(9\). The 95% critical value is approximately \(t^*=2.262\).
The 95% confidence interval is approximately \((0.08,\ 3.52)\) seconds. We are 95% confident that the population mean time in design A minus the population mean time in design B is between about 0.08 and 3.52 seconds. Positive differences correspond to faster times in design B, according to the definition of \(d\).
The interval estimates an average difference, not the change for every runner. Because the runners were randomly selected, the estimate can describe the population represented by that sample. The randomized order helps guard against order effects, but does not turn the sample into a random assignment of shoe designs to separate groups.
Independent Groups: Keep the Order and Goal Straight
With two independent groups, each unit contributes a response to just one group, and there is no design-based one-to-one pairing. Use the unpooled two-sample t procedure. For a test, define \(\mu_1-\mu_2\) and write the alternative to match the question. For an interval, keep the same group order when reporting the estimate and its endpoints.
Worked Example: A Two-Sample t Test in a Randomized Experiment
In a fictional experiment, 30 volunteers are randomly assigned to one of two audio settings while completing a concentration task. Each volunteer uses only one setting. The response is the number of correct items, a quantitative score. For setting 1, \(n_1=15\), \(\bar{x}_1=42.6\), and \(s_1=4.5\). For setting 2, \(n_2=15\), \(\bar{x}_2=39.1\), and \(s_2=5.0\). The score distributions in both groups are roughly symmetric with no apparent outliers. The researchers ask whether the population mean score is higher with setting 1. Use \(\alpha=0.05\).
State: Let \(\mu_1\) and \(\mu_2\) be the true mean scores for the populations represented by volunteers using settings 1 and 2, respectively. The hypotheses are \(H_0:\mu_1-\mu_2=0\) and \(H_a:\mu_1-\mu_2>0\).
Plan: Use an unpooled two-sample t test for \(\mu_1-\mu_2\), because the response is quantitative, the groups consist of different volunteers, and the question asks whether the means differ in a specified direction. Participants were randomly assigned to settings, each contributed one score, and the groups are independent. The roughly symmetric score distributions with no apparent outliers in each group support the t procedure. There is no 10% condition to check here: the prompt describes random assignment, not sampling without replacement from a stated finite population.
Do: The unpooled standard error and test statistic are
Using the calculator’s unpooled degrees of freedom, approximately \(27.7\), the right-tailed p-value is about \(0.027\), rounded. Since \(0.027<0.05\), reject \(H_0\).
Conclude: The data provide convincing evidence that the population mean concentration-task score is higher with setting 1 than with setting 2.
The experiment’s random assignment supports a cause-and-effect comparison between the settings for the participants. It does not, by itself, justify generalizing the result to all people, because the volunteers were not described as a random sample from that broader population.
Worked Example: A Two-Sample t Interval for a Mean Difference
A fictional city randomly selects 12 neighborhood gardens from 160 gardens using a new compost mix and 10 different gardens from 140 gardens using the usual mix. The response is weekly vegetable yield in kilograms. The new-mix group has \(\bar{x}_1=31.2\) kg and \(s_1=3.6\) kg; the usual-mix group has \(\bar{x}_2=28.0\) kg and \(s_2=4.0\) kg. Plots show roughly symmetric distributions with no apparent outliers in either group. The city wants a 95% confidence interval for the difference in population mean weekly yield, new mix minus usual mix.
Identify and check: These are two distinct, unlinked groups, so use an unpooled two-sample t interval for \(\mu_1-\mu_2\), where group 1 is gardens using the new mix and group 2 is gardens using the usual mix. Both groups were randomly sampled. The 10% conditions hold: \(12\leq0.10(160)=16\) and \(10\leq0.10(140)=14\). The observations are independent within and between groups, and each group’s roughly symmetric distribution with no apparent outliers supports the t procedure.
Calculate: The observed difference is \(31.2-28.0=3.2\) kg. The standard error is
The unpooled degrees of freedom are approximately \(18.4\), giving a 95% critical value of about \(2.10\). Thus,
We are 95% confident that the population mean weekly yield with the new mix minus the population mean weekly yield with the usual mix is between approximately \(-0.24\) and \(6.64\) kg. Since zero is in the interval, the data do not establish a clear difference in mean yield at this confidence level. The interval is an estimate of the difference, not a test conclusion that the population means are equal.
Common Mistakes in Mixed Practice
- Treating a benchmark as a second group: A target such as 500 mL is a fixed value, not a sample. One sample compared with that value calls for one-sample t inference.
- Calling data paired because there are two columns: Pairing comes from the study design. Identify the same-unit or deliberate matching link, then analyze one difference per pair.
- Using the wrong shape information: Check the differences for paired t, and check each group separately for a two-sample t procedure. For one-sample t, examine the individual observations.
- Choosing an interval when the question asks for evidence, or vice versa: “Is the mean higher?” asks for a test. “Estimate the mean difference” asks for an interval. The same design can support either goal.
- Omitting parameter order: State which group is group 1 and which is group 2, or define \(d\). Otherwise the sign and meaning of the estimate can be unclear.
- Claiming unsupported conditions: If the prompt does not describe random selection or assignment, independence, or data shape, say what information is missing. Do not write “all conditions are met” without evidence.
- Confusing random assignment with random sampling: Random assignment supports a treatment comparison; random sampling supports generalizing to a population. Do not claim one when the prompt only describes the other.
A complete setup is not a list of procedure names. It shows how the question, parameter, design, goal, and conditions fit together. For a test, the State–Plan–Do–Conclude structure from “Writing the Full Two-Sample t Test Solution” is a useful model. For an interval, define the parameter, justify the method, show the interval calculation, and interpret its endpoints in context.
Check Your Understanding
For each scenario, identify the population target, the procedure, and the condition evidence that should be checked. State whether the question calls for a test or an interval.
- A random sample of 18 transit buses from a fleet of 150 is measured for average battery range. The question asks whether the population mean is below a fixed 240-kilometer benchmark. What is the parameter, and what procedure fits?
- The same 12 students complete a typing task before and after practice. The question asks whether average speed increased. How should the difference be defined, and which distribution’s shape matters?
- Two separate groups of randomly assigned users try different navigation layouts, and the response is completion time. The question asks for an estimate of the mean time difference. What procedure and parameter should be used?
- A prompt says two groups are compared but does not explain whether observations are linked. What design information is needed before choosing paired or two-sample t?
- A test uses a random sample and specifies the sample size and population size but gives no information about shape or outliers. Which conditions can be checked, and what should you avoid asserting?