Confidence Interval for a Mean in Context

From a t Interval to a Contextual Answer

A confidence interval calculation is not complete when you have two endpoints. You also need to explain what those endpoints estimate, check that a one-sample t procedure is appropriate, and state the result in the language of the problem. In the previous tutorial, “Common Errors in Interpreting t Intervals,” we distinguished the population mean from individual values and reviewed what a confidence level describes. Here, we put those ideas to work in contextual examples.

Suppose a study measures a quantitative variable, such as hours of sleep or battery life, on a random sample. The goal is to estimate the population mean \(\mu\). Since \(\sigma\), the population standard deviation, is usually unknown, a one-sample t interval uses the sample standard deviation \(s\). As in “The Form of a One-Sample t Interval,” the interval is centered at \(\bar{x}\); its estimated standard error is \(s/\sqrt{n}\).

Formula: A one-sample t confidence interval for a population mean is \(\bar{x}\pm t^*(s/\sqrt{n})\). Equivalently, its endpoints are \(\bar{x}-t^*(s/\sqrt{n})\) and \(\bar{x}+t^*(s/\sqrt{n})\), with \(df=n-1\).

The confidence level determines the critical value \(t^*\), together with the degrees of freedom. The margin of error is \(t^*(s/\sqrt{n})\). Use the units of the measured variable for the standard error, margin of error, and both interval endpoints. The t interval procedure and the ways to find \(t^*\) were introduced in “Building a One-Sample t Interval by Hand” and “Finding t* Critical Values With a Table or invT.”

Conditions Before You Calculate

An interval can be calculated even when the sampling method or data shape does not justify it. Before using the formula, describe how the data were collected and check the conditions for inference. For sampling without replacement from a finite population, the 10% condition helps support treating observations as independent. With a small sample, the data should come from a population that is approximately Normal, or the sample data should show no strong skewness or outliers. For a larger sample, a t procedure is generally more robust to departures from Normality, but extreme skewness or outliers remain a concern.

Conditions: Check that the data come from a random sample or an appropriate randomized process; that observations are independent (for sampling without replacement, verify the 10% condition, \(n<0.10N\)); and that the population or sample shape is suitable for a t procedure. For a small sample, look for approximate Normality and no outliers. For a larger sample, check for severe skewness or outliers.

A condition check should say what supports the condition in the particular situation. For example, “the data are random” is not enough if the problem does not say how they were selected. Likewise, a sample size alone does not prove the population is Normal. Use information actually given, and be clear when an assumption is being made.

Worked Examples

Worked Example: Mean Sleep Time for Students

A fictional school researcher selects a simple random sample of 16 students from a school with more than 160 students. Students report their typical hours of sleep on a school night. The sample has \(\bar{x}=7.1\) hours and \(s=1.2\) hours. A plot of the sample data shows no strong skewness or outliers. Find and interpret a 95% confidence interval for the mean sleep time of all students at the school.

State. Let \(\mu\) be the true mean number of hours of sleep on a school night for all students at this school. We want to estimate this population mean using a one-sample t interval.

Plan and conditions. A t interval is appropriate because the population standard deviation is unknown and the sample standard deviation is provided. The 16 students were selected by simple random sampling. Because the sample was taken without replacement from more than 160 students, \(16<0.10N\), so the 10% condition is met and independence is reasonable. The sample has no strong skewness or outliers, supporting the t procedure for this sample of 16.

Do. The degrees of freedom are \(df=n-1=16-1=15\). For a 95% interval with 15 degrees of freedom, \(t^*\approx2.1314\). The estimated standard error is:

\frac{s}{\sqrt{n}} =\frac{1.2}{\sqrt{16}} =\frac{1.2}{4} =0.3\text{ hours}

The margin of error is \(t^*\) times the estimated standard error:

ME=2.1314(0.3)=0.63942\text{ hours}

Add and subtract this margin of error from the sample mean:

7.1\pm0.63942 \quad\Longrightarrow\quad (6.46058,\ 7.73942)\text{ hours}

Conclude. We are 95% confident that the true mean number of hours of sleep on a school night for all students at this school is between approximately 6.461 and 7.739 hours. This interval estimates the population mean; it does not say that 95% of individual students sleep between those endpoints.

Worked Example: Mean Battery Life

A fictional manufacturer randomly selects 12 rechargeable batteries from a large production run and measures how long each lasts under the same test conditions. The sample mean battery life is 9.6 hours, and the sample standard deviation is 0.8 hours. Assume the population distribution of battery life under these test conditions is approximately Normal, and that the production run contains more than 120 batteries. Construct and interpret a 90% confidence interval for the mean battery life.

State. Let \(\mu\) be the true mean battery life, in hours, for batteries from this production run when tested under the stated conditions.

Plan and conditions. Use a one-sample t interval to estimate \(\mu\), since \(\sigma\) is unknown. The batteries were selected randomly. Since \(12<0.10N\), the 10% condition is met, supporting independence when sampling without replacement. The assumed approximately Normal population distribution supports using a t interval with this small sample.

Do. The degrees of freedom are \(df=12-1=11\). For a 90% confidence interval with 11 degrees of freedom, \(t^*\approx1.7959\). The standard error is:

\frac{s}{\sqrt{n}} =\frac{0.8}{\sqrt{12}} \approx0.23094\text{ hours}

The margin of error is approximately \(1.7959(0.23094)=0.41474\) hours. Therefore, the interval is:

9.6\pm0.41474 \quad\Longrightarrow\quad (9.18526,\ 10.01474)\text{ hours}

Conclude. We are 90% confident that the true mean battery life for batteries from this production run, under the stated test conditions, is between approximately 9.185 and 10.015 hours. The 90% confidence level describes the long-run capture rate of this interval procedure if appropriate random samples were repeatedly selected and intervals constructed in the same way.

Worked Example: Mean Commute Time to School

A fictional school surveys a simple random sample of 25 students about their one-way commute time, in minutes. The sample mean is 18.4 minutes, and the sample standard deviation is 4.5 minutes. The school has more than 250 students, and a graph of the sample data shows no extreme outliers or strong skewness. Find and interpret a 95% confidence interval for the mean one-way commute time of all students at the school.

State. Let \(\mu\) be the true mean one-way commute time, in minutes, for all students at this school.

Plan and conditions. Use a one-sample t interval because the goal is to estimate a population mean and \(\sigma\) is unknown. The sample was selected randomly. Since \(25<0.10N\), the 10% condition is satisfied, so observations can be treated as independent. The sample has no extreme outliers or strong skewness; this supports a t procedure with \(n=25\).

Do. The degrees of freedom are \(df=25-1=24\). For a 95% confidence interval with 24 degrees of freedom, \(t^*\approx2.0639\). Calculate the standard error and margin of error:

\frac{s}{\sqrt{n}} =\frac{4.5}{\sqrt{25}} =\frac{4.5}{5} =0.9\text{ minutes}

ME=2.0639(0.9)=1.85751\text{ minutes}

The interval endpoints are:

18.4\pm1.85751 \quad\Longrightarrow\quad (16.54249,\ 20.25751)\text{ minutes}

Conclude. We are 95% confident that the true mean one-way commute time for all students at this school is between approximately 16.542 and 20.258 minutes. The interval gives a plausible range for the population mean, not for the commute time of every student.

Making the Answer Complete

Each example follows the same logic: identify the population mean, justify the procedure, show how the interval was calculated, and interpret the result in context. The arithmetic is connected to the situation at every stage. For example, a standard error of 0.3 hours is a measure of estimated variability in sample means, while \(s=1.2\) hours describes variability among individual sleep-time reports. Keeping these quantities distinct helps prevent unit and interpretation errors.

When reporting endpoints, round to a sensible level of precision and keep the units. Rounding the endpoints does not change the method, but it can make a result easier to read. Avoid reporting many calculator digits as though the data support that level of precision. If the calculated interval has a negative lower endpoint for a quantity that cannot be negative, do not change the interval silently; report the calculation and consider whether the procedure or model is suitable for the context.

A t interval describes uncertainty in estimating a mean from a sample. It does not correct for a biased sampling method, and a large sample does not make a nonrandom sample representative by itself. The random-sampling and independence checks are therefore part of the statistical conclusion, not optional decoration.

Common Mistakes and AP Exam Tips

Leaving the parameter vague. A response that says “the mean is between the endpoints” should identify the population and measured quantity. Define \(\mu\) in context before calculating.
Skipping a condition. State how the sample was selected, address independence and the 10% condition when needed, and explain why the sample or population shape supports a t procedure.
Using the wrong spread. For a one-sample t interval, use \(s/\sqrt{n}\) as the estimated standard error, not \(s\) by itself. The standard deviation \(s\) describes individual sample values; the standard error estimates the variability of sample means.
Using the wrong degrees of freedom or critical value. For this one-sample t interval, calculate \(df=n-1\), then use the \(t^*\) matching both the confidence level and those degrees of freedom.
Interpreting the interval as a range for individuals. Name the true population mean as the target. As emphasized in “Common Errors in Interpreting t Intervals,” the interval does not say that a specified percentage of individual values falls between its endpoints.
Omitting units or context in the conclusion. A full-credit interpretation identifies the confidence level, both endpoints, the population mean, the population, and the units.

A useful final check is to read your conclusion aloud: Would a reader know what is being averaged, for whom or what, and in what units? Does the sentence say the interval estimates the true population mean rather than individual observations? If so, the numerical work and the contextual interpretation are working together.

Key takeaway: A complete contextual t-interval solution defines the population mean, checks the conditions, calculates the interval using \(df=n-1\), and interprets both endpoints with the confidence level, population, quantity, and units.

Check Your Understanding

Use the one-sample t interval ideas in these situations. Include context and units in your responses.

A random sample of 20 hikers has a mean walk time of 42 minutes and a sample standard deviation of 10 minutes. What is the parameter of interest, and what information about the population and sample shape would you want before using a t interval?
For a sample of 15 devices, explain how to find the degrees of freedom for a t interval and identify what else is needed to determine \(t^*\).
A 95% t interval for mean weekly practice time is \((4.2,\ 6.8)\) hours. Write an interpretation naming the population and the quantity being estimated.
Why is \(s\), rather than \(\sigma\), used in the standard error for a one-sample t interval?
A student says, “95% of the batteries have lifetimes between the endpoints of this 95% confidence interval.” Explain the specific error in this statement.