What a t Interval Describes
A one-sample t interval estimates a population mean. Its endpoints are calculated from sample data, but the target is the fixed, unknown population mean \(\mu\). In “Interpreting a Confidence Interval for a Mean” and “Interpreting the Confidence Level for a Mean,” we saw how to state what an interval and its confidence level mean. Here, we focus on errors that can make an otherwise correct calculation sound incorrect.
Two common mistakes are saying that 95% of individuals fall in a 95% confidence interval, and saying that there is a 95% probability that the fixed population mean is in the particular interval after it has been calculated. Neither statement describes the standard AP Statistics interpretation. The first confuses a mean with individual values. The second treats \(\mu\) as random instead of recognizing that the interval varies from sample to sample.
A t interval has the form \(\bar{x}\pm t^*(s/\sqrt{n})\). Its center is the sample mean \(\bar{x}\), and its width reflects the estimated variability of sample means. It is not designed to describe where most individual observations fall. Individual observations vary around the population mean, and their spread is described by the population distribution—not by the confidence interval for the mean.
Common Interpretations to Correct
The statement “95% of individuals are in this interval” is wrong because the interval estimates \(\mu\), not the range containing individual values. Some individuals may fall inside it and some outside it; the interval alone does not tell us what proportion falls within those endpoints. A different kind of interval would be needed to estimate or predict a range for individual values.
The statement “there is a 95% chance that \(\mu\) is in this particular interval” also misses the AP interpretation. In the standard confidence-interval framework, \(\mu\) is a fixed population value. Before collecting a sample, the interval’s endpoints are not yet known and can vary from sample to sample. After the sample is collected, the endpoints are fixed too. The confidence level describes how the interval-producing method performs over many repetitions—not a probability assigned to \(\mu\) after calculating one interval.
The word “confident” refers to the method’s long-run capture rate. If we repeatedly took appropriate random samples and built an interval the same way each time, about 95% of the intervals from a 95% procedure would contain the fixed \(\mu\). The other intervals would miss it. We generally do not know which individual interval is among the successful ones.
Ask whether the interval estimates a population mean or describes individual values. A one-sample t interval estimates \(\mu\).
The population mean does not change from sample to sample. The sample mean and the interval endpoints can change.
Describe the long-run capture rate of the interval procedure, then state the interval’s endpoints, target population, and units.
Worked Examples
Worked Example: Does the Interval Contain 95% of Students?
A fictional school district takes a random sample of 25 high school students to estimate the mean daily recreational screen time for all its high school students. The sample has \(\bar{x}=4.8\) hours and \(s=2.0\) hours. The sample was selected from a district population much larger than 250 students. The data show no strong skewness or outliers. Find a 95% t interval and assess this claim: “95% of students in the district have daily recreational screen time between the interval’s endpoints.”
State. Let \(\mu\) be the true mean daily recreational screen time, in hours, for all high school students in the district. The claim is about individual students, while the interval estimates \(\mu\).
Plan and conditions. Use a one-sample t interval because the population standard deviation is unknown and the sample standard deviation is available. The sample is random. The observations are independent; sampling without replacement from a population much larger than 250 means \(25<0.10N\), so the 10% condition is met. With \(n=25\), the stated absence of strong skewness or outliers supports using the t procedure.
Do. The degrees of freedom are \(df=25-1=24\). For a 95% interval with 24 degrees of freedom, \(t^*\approx2.0639\). The estimated standard error and margin of error are:
The interval is:
Conclude. We are 95% confident that the true mean daily recreational screen time for all high school students in the district is between approximately 3.974 and 5.626 hours. The claim about 95% of individual students is not supported by this interval: it estimates the population mean, not the percentage of individual students whose values lie between those endpoints.
Worked Example: Is the Population Mean Random?
A fictional repair center takes a random sample of 10 service requests to estimate the mean time, in hours, to resolve all requests. The sample summary is \(n=10\), \(\bar{x}=12.4\) hours, and \(s=3.0\) hours. Assume the population distribution is approximately Normal, and the center handles far more than 100 requests. A student says, “There is a 95% probability that the true mean resolution time is in my interval.” Find the interval and revise the interpretation.
State. Let \(\mu\) be the true mean resolution time for all service requests handled by this repair center. The goal is to estimate this fixed population mean.
Plan and conditions. Use a one-sample t interval. The sample is random, and \(10<0.10N\) because the center handles far more than 100 requests, supporting the 10% condition and independence. The population is assumed approximately Normal, which supports the t procedure for this small sample.
Do. The degrees of freedom are \(df=10-1=9\), and for a 95% interval \(t^*\approx2.2622\). The estimated standard error and margin of error are:
Therefore:
Conclude. We are 95% confident that the true mean resolution time for all service requests handled by this repair center is between approximately 10.254 and 14.546 hours. The student’s probability statement is not the standard interpretation: \(\mu\) is fixed, not random. The 95% describes the long-run success rate of the interval procedure if appropriate random samples were repeatedly taken and intervals constructed in the same way.
Worked Example: What 95% Confidence Means Across Repeated Samples
A fictional community center takes a random sample of 36 members to estimate the mean number of hours members spend at the center each week. The sample summary is \(\bar{x}=7.4\) hours and \(s=1.8\) hours. Assume the center has more than 360 members, and the sample data have no extreme outliers. Use a 95% t interval to show how to interpret the result and the confidence level.
State. Let \(\mu\) be the true mean number of hours all community center members spend there each week.
Plan and conditions. Use a one-sample t interval because \(\sigma\) is unknown. The sample is random, the population size exceeds 360 so \(36<0.10N\), and the 10% condition supports treating observations as independent. With \(n=36\) and no extreme outliers, the sample is sufficiently large and well behaved for the t procedure.
Do. Here, \(df=36-1=35\), and the 95% critical value is \(t^*\approx2.0301\). The estimated standard error is \(1.8/\sqrt{36}=0.3\) hours. The margin of error is \(2.0301(0.3)=0.60903\) hours. Thus:
Conclude. We are 95% confident that the true mean weekly time at the center for all its members is between approximately 6.791 and 8.009 hours. If we repeatedly took random samples of 36 members and constructed intervals using this same 95% t procedure, about 95% of those intervals would capture the fixed \(\mu\). That statement is about the procedure across repetitions; it does not say that 95% of members spend between 6.791 and 8.009 hours at the center.
Common Mistakes and AP Exam Tips
- Describing individuals instead of the mean. “95% of students are between the endpoints” is not an interpretation of a t interval for \(\mu\). A full-credit response names the population mean and does not claim a percentage of individual values.
- Calling \(\mu\) random. Avoid “there is a 95% probability that \(\mu\) is in this interval.” Say that we are 95% confident the interval captures the fixed population mean, or explain that about 95% of intervals from repeated appropriate samples would capture it.
- Confusing the sample mean and population mean. The sample mean \(\bar{x}\) is the interval’s center and is calculated from the sample. The population mean \(\mu\) is the unknown target being estimated. They are not interchangeable.
- Leaving out context or units. “We are 95% confident the mean is between 10.254 and 14.546” is incomplete if it does not identify what is being averaged and the units. State the population, quantity, and units.
- Assuming correct arithmetic guarantees a correct interpretation. The conditions and calculations matter, but so does saying what the interval actually estimates. A calculator cannot fix a mismatch between the interval’s target and the claim being made.
On an AP response, first name \(\mu\) in context. Then give the endpoints with units and state that you are confident the interval captures that true population mean. If asked what the confidence level means, describe the long-run proportion of intervals from repeated appropriate samples that would contain \(\mu\). Keep the individual values, sample mean, and population mean distinct.
Check Your Understanding
For each item, identify the target of the interval and correct any mistaken interpretation.
- A 95% t interval for mean weekly exercise time is \((2.1,\ 3.4)\) hours. Is it correct to say that 95% of people exercise between 2.1 and 3.4 hours per week? Explain.
- Why is “there is a 90% probability that \(\mu\) is in this particular 90% interval” not the standard AP interpretation?
- Write a contextual interpretation for a 95% interval from 18.2 to 21.6 kilograms estimating the mean mass of packages produced by a factory.
- In repeated sampling, which can vary: the population mean, the sample mean, or the interval endpoints? Explain.
- What does the 95% confidence level say about intervals constructed from many repetitions of the same appropriate procedure?