A Four-Step Structure for a Full Response
A one-sample t interval answer should do more than report two endpoints. A reader should be able to tell what population mean is being estimated, why the procedure is appropriate, how the interval was calculated, and what the interval means in context. The previous tutorial, “Confidence Interval for a Mean in Context,” practiced those ingredients. Here, the goal is to organize them in the State, Plan, Do, Conclude structure often used for free-response answers.
Treat the four steps as a connected argument, not four unrelated labels. State identifies the parameter and goal. Plan names the procedure and addresses its conditions. Do shows the calculation clearly enough to follow. Conclude answers the original question by interpreting the interval for the population in the problem.
Define \(\mu\) in context and say that you are estimating it with a confidence interval.
Name a one-sample t interval and explain how the sampling, independence, and shape conditions are met.
Find \(df=n-1\), identify \(t^*\), calculate the estimated standard error and margin of error, and report the interval.
State the confidence level and interval endpoints, then identify the population mean, population, quantity, and units.
This order helps expose missing reasoning. For instance, a correct interval without a condition check is not a complete inference response; a careful condition check without an interpretation does not answer what the estimate says. As in “Building a One-Sample t Interval by Hand,” the formula is \(\bar{x}\pm t^*(s/\sqrt{n})\), with \(df=n-1\). The focus here is on making every part of a response visible.
Worked Examples
Worked Example: Mean Weekly Harvest From Garden Plots
A fictional community garden coordinator selects a simple random sample of 25 plots from a garden network with more than 250 plots. For one week, the coordinator records the harvest mass from each selected plot. The sample mean is 32.6 kilograms, and the sample standard deviation is 6.0 kilograms. A graph of the sample data shows no strong skewness or outliers. Write a complete four-step solution for a 95% confidence interval for the mean weekly harvest mass across the network’s plots.
State. Let \(\mu\) be the true mean weekly harvest mass, in kilograms, for all plots in this community garden network. We will estimate \(\mu\) with a 95% confidence interval.
Plan. Use a one-sample t interval because the goal is to estimate a population mean and the population standard deviation \(\sigma\) is unknown. The plots were selected by simple random sampling. Because the sample was drawn without replacement from more than 250 plots, \(25<0.10N\), so the 10% condition is met and treating observations as independent is reasonable. The sample graph shows no strong skewness or outliers, supporting the use of a t interval.
Do. There are \(n=25\) plots, so \(df=n-1=24\). For a 95% confidence interval with 24 degrees of freedom, \(t^*\approx2.0639\). The estimated standard error and margin of error are:
Adding and subtracting the margin of error from the sample mean gives:
Conclude. We are 95% confident that the true mean weekly harvest mass for all plots in this community garden network is between approximately 30.123 and 35.077 kilograms. The interval estimates the network’s population mean weekly harvest mass, not the harvest from every individual plot.
Worked Example: Mean Nitrate Concentration in Wells
A fictional environmental team randomly selects 10 wells from a region with more than 100 wells and measures nitrate concentration in each well. The sample mean is 5.8 milligrams per liter, and the sample standard deviation is 2.4 milligrams per liter. The region’s well measurements are assumed to come from an approximately Normal population. Construct and interpret a 90% confidence interval for the mean nitrate concentration in the region’s wells.
State. Let \(\mu\) be the true mean nitrate concentration, in milligrams per liter, among wells in this region. We want a 90% confidence interval for \(\mu\).
Plan. Use a one-sample t interval to estimate the population mean because \(\sigma\) is unknown. The wells were randomly selected. Since \(10<0.10N\), the 10% condition is met, supporting independence when sampling without replacement. The approximately Normal population model supports using a t interval with this small sample.
Do. The degrees of freedom are \(df=10-1=9\). For a 90% confidence interval with 9 degrees of freedom, \(t^*\approx1.8331\). The estimated standard error is:
The margin of error is approximately \(1.8331(0.75895)=1.39123\) milligrams per liter. Thus, the interval is:
Conclude. We are 90% confident that the true mean nitrate concentration among wells in this region is between approximately 4.409 and 7.191 milligrams per liter. This interpretation refers to the population mean concentration, not to a range that contains a specified percentage of individual wells.
Worked Example: Mean Battery-Charging Time for Scooters
A fictional repair center randomly selects 40 electric scooters from a large fleet and measures the time, in minutes, each takes to charge under the same conditions. The sample mean is 71.3 minutes, and the sample standard deviation is 8.0 minutes. A graph shows no extreme outliers or strong skewness, and the fleet has more than 400 scooters. Write a full four-step solution for a 95% confidence interval for the mean charging time.
State. Let \(\mu\) be the true mean charging time, in minutes, for all scooters in this fleet under the stated charging conditions. We will estimate this mean with a 95% confidence interval.
Plan. Use a one-sample t interval because the parameter is a population mean and \(\sigma\) is unknown. The scooters were randomly selected. Since \(40<0.10N\), the 10% condition is met, making it reasonable to treat the observations as independent. The sample shows no extreme outliers or strong skewness; with \(n=40\), this supports using a t procedure.
Do. The degrees of freedom are \(df=40-1=39\). For a 95% interval with 39 degrees of freedom, \(t^*\approx2.0227\). Calculate the standard error and margin of error:
The interval endpoints are:
Conclude. We are 95% confident that the true mean charging time for all scooters in this fleet, under the stated conditions, is between approximately 68.741 and 73.859 minutes.
What Makes the Four Steps Work?
The examples use the same sequence, but the Plan is specific to each situation. In the garden example, the sample was random, the 10% condition could be checked using the network size, and the sample graph supported the shape condition. In the well example, an approximately Normal population was stated, which matters because the sample was small. In the scooter example, the sample was larger and the graph showed no extreme outliers or strong skewness. A strong response does not copy a generic sentence; it connects each condition to the facts given.
In the Do step, include enough calculation for the reader to verify the interval. At minimum, show \(df=n-1\), the critical value used, the estimated standard error, and the interval calculation. The margin of error may be shown as a separate line, which makes clear how far each endpoint is from \(\bar{x}\). Keep the units attached to quantities such as the standard error and margin of error.
If you use a calculator’s TInterval function, the calculator can produce the endpoints, but your written response still needs the State, Plan, and Conclude reasoning. In a free-response solution, calculator output does not explain why the procedure fits the data or what population mean the result estimates. When starting with summary statistics, enter \(\bar{x}\), \(s\), \(n\), and the confidence level carefully; as covered in “Confidence Intervals From Summary Statistics Versus Raw Data,” use the sample standard deviation.
Common Mistakes and AP Exam Tips
- Writing only “let \(\mu\) be the mean.” This does not say what is averaged or whose mean is being estimated. Define the quantity, population, and units in the State step.
- Calling the procedure a t interval without checking conditions. In the Plan step, describe the sampling method, address independence and the 10% condition when appropriate, and explain the evidence about shape.
- Giving calculator endpoints with no visible method. Include the degrees of freedom, \(t^*\), standard error, and margin of error or equivalent endpoint calculation so the Do step is understandable.
- Mixing up the sample and population quantities. The sample mean \(\bar{x}\) is calculated from the observed data; \(\mu\) is the fixed population mean the interval estimates. Name \(\mu\) in the conclusion.
- Turning the conclusion into a claim about individuals. A confidence interval for a mean is not an interval for individual measurements. The conclusion should describe the true population mean in context.
- Leaving out units or confidence level. A full-credit conclusion states the confidence level, both endpoints, the population mean, the population, the measured quantity, and units.
A useful final audit is to check the response in order: Is the parameter defined? Does the Plan justify the t interval using facts from the problem? Can the reader follow the calculation? Does the last sentence answer the question about the population mean? If any answer is no, revise that step rather than adding an unrelated sentence at the end.
Check Your Understanding
For each prompt, focus on organizing the response rather than merely naming a formula.
- A simple random sample of 18 community volunteers has a mean weekly recycling mass of 4.2 kilograms and a sample standard deviation of 1.5 kilograms. What information belongs in the State step, and what additional facts would you need to complete the Plan step?
- For a one-sample t interval based on \(n=14\), find the degrees of freedom. What two pieces of information determine the positive critical value \(t^*\)?
- A student has correctly calculated a 95% interval for a population mean but concludes, “95% of the individual measurements are between these endpoints.” Identify the error and describe what the conclusion should refer to.
- Why is it not enough to write “conditions are met” in the Plan step? Give two condition checks that should be tied to facts from the problem.
- A complete interval response has a parameter definition, condition check, and calculation but no contextual final sentence. Which step is missing, and what should that step communicate?