Week 11 · Chapter

Chapter 11 — Confidence Intervals for a Mean

Introduction to Statistics Free sample edition

The module's primary reading · ~45–55 minutes · Objective 6 (construct and correctly interpret confidence intervals for a population mean and a population proportion, including the meaning of the confidence level and the margin of error) · Read it first this week; the lecture, the tutorial, the quiz, and the Data Lab all build on it.

In plain language

When you measure a sample, its average is a good guess for the whole group, but it is never exactly right. This week you learn to report a confidence interval (a range of believable values for the group's true average, with a stated level of trust) instead of a single number. You will see why the range is built the way it is and how to compute it by hand and with a spreadsheet. You will also learn to say what the range means in plain words, and what it does not mean. Getting that sentence right is the main skill of the week.

Why you already care about this

A bag of coffee says 340 g on the label. A weather app says tomorrow's high will be 24 degrees. A clinic's sign says the average wait is 15 minutes. Each one is a single number, and each one hides a wobble. Nobody thinks every bag weighs exactly 340 g, or that tomorrow will hit 24 degrees on the dot.

The honest version of each number comes with a range: "about 15 minutes, give or take 3". The range tells you how much to trust the number. It is the difference between planning your afternoon around a wait time and getting caught by surprise.

Field biologists face the same problem. A research team in Antarctica weighed 151 Adelie penguins, and the birds averaged 3,700.66 g. That is the team's best single guess for the average Adelie penguin. But a different 151 birds would have averaged something a little different. Last week you learned how much a sample mean wobbles. This week you turn that wobble into an honest range.

By the end of this chapter you will be able to:

tell a point estimate from an interval estimate, and explain why a range is more honest than one number;
explain why an interval built from the sample mean captures the true mean for about 95% of samples;
find the degrees of freedom and the critical value t* for a sample, and explain why t* is bigger for small samples;
check the conditions for a one-sample t interval before you compute it;
compute a 95% confidence interval for a mean by hand and with a spreadsheet;
interpret the interval in one correct sentence;
predict how the confidence level, the spread, and the sample size change the margin of error;
explain what "95% confident" does and does not mean.

Carry this sentence with you: the interval is about the mean — not about single birds.

Where we are

Already covered: Weeks 1 to 3 described one variable, Week 4 related two, Weeks 5 to 7 built probability and the binomial, Week 8 added the z-score, Week 9 the normal model, and Week 10 the sampling distribution of the mean.
This week: confidence intervals for a mean: point and interval estimates, the t distribution, the conditions, the one-sample t interval, the margin of error, and what a confidence level means.
How they connect: last week showed how far a sample mean typically lands from the true mean; this week you run that idea backward to estimate the true mean from one sample.
Where we're going next: this week opens the inference unit. Week 12 builds the same kind of interval for a proportion, and a new question waits after that: how do you test a claim?

Three earlier weeks matter most here. Week 1 gave you the line between a sample (the part you measure) and a population (the whole group you ask about), and between a statistic and a parameter. Week 9 gave you the numbers 1.645, 1.960, and 2.576, the z-scores that hold the middle 90%, 95%, and 99% of a bell. Week 10 gave you the standard error, how much the sample mean typically wobbles from sample to sample, and the Central Limit Theorem, which says the sample mean follows a roughly normal model when the sample is large. This week those three ideas become one tool.

1. Point estimates: one number and its wobble

Start with the penguins. A team of field biologists working with the Palmer Station research program weighed Adelie penguins on three islands. Of the 152 Adelie birds in the study, 151 have a recorded body mass. Those 151 birds are a sample. The population is every Adelie penguin like them, and nobody can weigh all of those.

The team wants one number: the mean body mass of the population. Call it μ (read "mu"). It is a parameter, a fixed number that nobody knows. What the team does know is x̄ (read "x bar", the sample mean), a statistic computed from the birds they weighed.

A point estimate is a single number computed from a sample and used as the best guess for a population parameter; the sample mean x̄ is the point estimate of the population mean μ.

Worked example — the Adelie point estimate. The question: what is the mean body mass μ of Adelie penguins like the ones weighed in the Palmer field study?

Name the sample. The 151 measured Adelie penguins are a sample, and their mean is a statistic.
Compute the statistic. x̄ (the sample mean) is 3,700.66 g for the 151 measured Adelie penguins.
Name the spread of single birds. s (the sample standard deviation) is 458.57 g for the 151 masses. Single birds vary a lot.

In words: the team's best single guess for the average Adelie penguin is 3,700.66 g. A team that reports only that number gives a reader no way to tell whether it is close to μ or far off.

Why worry? Week 10 showed that a sample mean changes from sample to sample. If another team weighed a different 151 Adelie birds, their x̄ would land somewhere else, maybe 3,680 g, maybe 3,725 g. One number hides that wobble.

Misconception check — "the sample mean is the true mean". Students often treat x̄ as if it were μ, because it is the only number on the page. It is tempting: 3,700.66 g looks precise to the hundredth of a gram. But x̄ describes the 151 birds that were weighed, and μ describes the whole population. The cure: say "estimate" every time you report x̄, and ask how far off it might be.

Memory hook: one number is a guess; a range is a guess with its wobble attached.

2. Interval estimates: turn the wobble into a range

How do you attach a wobble to a guess? Borrow last week's picture, and then flip it.

Take a fair die, a case where you know the truth. One roll has mean μ = 3.5 points. Now average 30 rolls, write the average down, and repeat many times. Week 10 showed that those averages pile up in a bell around 3.5. Their standard error is the given input 0.3118 points from Week 10, so averages of 30 rolls typically land within about a third of a point of 3.5.

Week 9 told you how wide the middle 95% of a bell is: 1.960 standard deviations each way. Put those together.

Worked example — the die interval. The die is the one place in this chapter where you know μ, so you can watch the method work.

Find the reach. Multiply 1.96 by the given input 0.3118: 1.96 × 0.3118 = 0.61 points, rounded to two decimals.
Find the band. About 95% of 30-roll averages land between 3.5 − 0.61 = 2.89 and 3.5 + 0.61 = 4.11 points.
Flip it. Whenever an average lands in that band, the interval x̄ ± 0.61 points built around it reaches back to 3.5. So about 95% of such intervals capture μ, and about 5% miss it.
A capture. Suppose one 30-roll sample averages 3.20 points. Its interval is 3.20 − 0.61 = 2.59 to 3.20 + 0.61 = 3.81 points, which contains 3.5.
A miss. Suppose another 30-roll sample averages 4.20 points. Its interval is 4.20 − 0.61 = 3.59 to 4.20 + 0.61 = 4.81 points, which does not contain 3.5.

In words: about 1 interval in 20 built this way misses the true mean. Nothing about the miss looks different from the inside. The 4.20 sample does not wave a flag; it happens to be one of the unlucky ones.

That flip is the whole logic of an interval estimate. You cannot see μ, but you know how far x̄ usually lands from it. So you build a range around x̄ that reaches that far.

An interval estimate is a range of plausible values for a population parameter, built as a point estimate plus or minus a margin of error, so that it shows both the guess and how far off the guess might be.
A margin of error is the distance from the point estimate to either end of a confidence interval, equal to the critical value times the standard error; it measures sampling variability only, not bias.
A confidence interval is an interval estimate built by a method that captures the population parameter in a stated share of all possible samples, such as 95%; it is reported as an estimate plus or minus a margin of error.

The ± sign is read "plus or minus". The interval x̄ ± 0.61 runs from x̄ minus 0.61 up to x̄ plus 0.61.

Misconception check — "the interval is centered on the true mean". Students often draw the interval around μ, because the band in step 2 was centered on 3.5. It is tempting since the two pictures look alike. But in real life you never know μ, so you cannot center anything on it. The cure: the band around μ is the theory; the interval around x̄ is what you actually build. Every confidence interval is centered on the sample's own estimate.

Pause and reflect — choose one. Pick either of the 2 prompts; they count the same.

(a) Think of a number you were given recently as a single value, such as a delivery time, a price estimate, or a battery life. What range would have made it more honest, and how would the range change what you did?
(b) A friend says, "If the interval can miss, why bother building it?" Write two or three sentences that answer your friend using the die example.

Memory hook: the band sits around the true mean; the interval sits around your estimate.

3. The t distribution: pay for an unknown spread

The die was easy because you know σ (read "sigma", the population standard deviation) for a fair die: about 1.71 points. You can list every face, so you can compute it.

For the penguins, you do not know σ. Nobody has weighed every Adelie penguin. So you plug in s (the sample standard deviation), 458.57 g, in its place. That swap is reasonable, but it adds a second guess on top of the first. Now x̄ wobbles, and so does s.

In 1908 a statistician who signed his work "Student" worked out what that extra guess does. What he found is a new family of curves.

The t distribution is a family of symmetric, bell-shaped curves centered at 0 with heavier tails than the standard normal distribution, used for a mean when the sample standard deviation s stands in for σ; it grows closer to the standard normal as the degrees of freedom grow.

"Heavier tails" means more area out at the edges. Picture the standard normal bell from Week 9, then press its peak down a little and let the extra area spread into the tails. Values far from 0 are a bit more common on a t curve. That is the price of guessing σ: you have to reach a little farther to catch the same 95%.

Which t curve do you use? It depends on how much information s is built on.

The degrees of freedom are the number, written df, that picks which t distribution to use; for one sample of size n it is df = n − 1.

Worked example — degrees of freedom for the penguins. n (the sample size) is 151, so df = 151 − 1 = 150. The penguin interval uses the t curve with 150 degrees of freedom.

With 150 degrees of freedom, the t curve is very close to the standard normal. With 9 degrees of freedom, from a sample of 10, it is noticeably flatter, with fatter tails. A researcher who weighs only 10 birds has to accept a wider interval for the same confidence.

Misconception check — "the data follow a t distribution". Students often think the t distribution describes the penguin masses themselves. It is tempting, because the t curve replaces the normal curve in the formula. But the t curve describes how the sample mean, measured in standard errors, varies from sample to sample when s stands in for σ. The cure: say "the t distribution describes the standardized sample mean", never "the data are t-distributed".

Memory hook: guess σ, and pay with heavier tails — not with a new formula.

4. The critical value: read it off the ladder for your sample

To build a 95% interval, you need the number of standard errors that holds the middle 95% of the right t curve. That number has a name.

A critical value is the number of standard errors a margin of error reaches for a chosen confidence level, read from the standard normal distribution (z*) or a t distribution (t*); for 95%, z* is 1.960.

You read z* as "z star" and t* as "t star". For a normal curve, Week 9 already gave you the critical values 1.645 for 90%, 1.960 for 95%, and 2.576 for 99%. For a t curve, the critical value depends on the degrees of freedom.

Here is the ladder of 95% critical values the course uses.

Critical values for 95% confidence by degrees of freedom, with the normal critical value for comparison
Sample size	Degrees of freedom	Critical value for 95%
10	9	t* is 2.262
25	24	t* is 2.064
151	150	t* is 1.976
very large	very large	z* is 1.960

Read the table from top to bottom. As the sample grows, t* shrinks toward z*. A sample of 10 pays a lot for its unknown σ. A sample of 151 barely notices: the critical value t* for 95% with df = 150 is 1.976, a hair above 1.960.

Worked example — how much wider is a small sample? Compare the critical values for 10 birds and for a very large sample.

Write the two critical values. With 9 degrees of freedom the critical value is 2.262; for the normal curve the critical value is 1.960.
Compare them. 2.262 ⁄ 1.960 ≈ 1.15.

In words: with only 10 birds, the interval reaches about 15% farther than a normal curve would, to pay for guessing σ. With 151 birds, the extra reach is under 1%.

In practice (your tools). In Google Sheets or Excel, =T.INV.2T(0.05, 150) returns the critical value for 95% with 150 degrees of freedom: 1.976, rounded to three decimals. The first input is the share left in the two tails together, so 0.05 for 95%, 0.10 for 90%, and 0.01 for 99%. The second input is the degrees of freedom. Try =T.INV.2T(0.05, 9) and =T.INV.2T(0.05, 24); they return 2.262 and 2.064. For the normal critical value, =NORM.S.INV(0.975) returns 1.960.

Misconception check — "df equals the sample size". Students often look up t* with 151 degrees of freedom for a sample of 151. It is tempting, because the sample size is the number right in front of you. But one sample's degrees of freedom are the sample size minus 1. For large samples the slip barely changes t*, but for a sample of 10 it does. The cure: write "df = n − 1" on the page before you look anything up.

Memory hook: small samples reach farther; big samples barely notice.

5. Conditions: check before you compute

A formula will happily turn any list of numbers into an interval. Whether the interval means anything depends on how the numbers were collected. So you check three conditions out loud, every time, before you compute.

Random or representative sample. The data should come from a random sample, or from a process you can defend as representative of the population.
Independence. One observation should not influence another.
Shape: a roughly normal population, or a large enough sample. For a small sample, the data should look roughly symmetric with no extreme outliers. For a large sample, the Central Limit Theorem takes care of shape. The course's rule of thumb is about 30 for a moderately skewed population; it is a working rule, not a law.

Worked example — conditions for the penguins. Check each one in words.

Sample. The penguins were weighed in an observational field study, not drawn at random, so you treat them as representative of Adelie penguins in that region and say so.
Independence. Each bird is weighed separately, so one mass does not set another.
Shape. A sample of 151 is far above the rule of thumb of about 30, so the sampling distribution of x̄ is approximately normal.

In words: all three conditions are reasonable, with one caveat to state plainly. The birds were not drawn at random, so the interval speaks for Adelie penguins like the ones the team found, in the places and years they worked.

Honesty note: this caveat is a real limitation of the week's own example. If the team mostly weighed birds near easy landing spots, and those birds differ from the rest, no formula can repair that. A reader deserves to hear it.

Misconception check — "a big sample fixes a bad sample". Students often think 151 birds is so many that the sampling method stops mattering. It is tempting, because a large n does fix the shape condition. But a large sample only shrinks random wobble; it does nothing about bias. You met this trap in Week 1. The cure: check the sample condition separately from the shape condition, and never let a big n answer a question about how the data were collected.

Memory hook: conditions first, arithmetic second — not the other way around.

6. The one-sample t interval: build it by hand and by spreadsheet

Now put the pieces together. When σ is unknown, the interval for a population mean has one standard form.

A one-sample t interval is the confidence interval for a population mean when σ is unknown, x̄ ± t* · s ⁄ √n, used after checking for a random or representative sample, independent observations, and a roughly normal population or a large enough sample.

Read it from the inside out. First, s ⁄ √n is the standard error, SE: how much the sample mean typically wobbles. Second, t* times the standard error is the margin of error. Third, x̄ plus or minus that margin gives the two ends.

Worked example — the 95% interval for the Adelie mean. Use the outline's values: 151 birds, x̄ of 3,700.66 g, and s of 458.57 g.

Degrees of freedom. df = 151 − 1 = 150.
Critical value. The critical value t* for 95% with df = 150 is 1.976.
Standard error. SE = s ⁄ √n, which here is 458.57 ⁄ √151 ≈ 37.32 g, because √151 ≈ 12.29. You divide by the square root of the sample size, not by the sample size itself.
Margin of error. Multiply the critical value 1.976 by SE = 37.32 g. The margin of error is about 73.74 g, rounded to two decimals.
Step down and up. 3,700.66 − 73.74 = 3,626.92 g and 3,700.66 + 73.74 = 3,774.40 g.
Report. The 95% t interval runs from 3,626.92 g to 3,774.40 g.

In words: the true mean mass of Adelie penguins like these is pinned down to a window about 147 g wide. A biologist who wants to compare Adelie birds with another species can now decide whether the gap is bigger than this wobble.

Check the width. 3,774.40 − 3,626.92 = 147.48 g, which is 2 × 73.74 g. The interval is always twice the margin of error wide, because it reaches the same distance on each side.

In practice (your tools). In Google Sheets or Excel, put the 151 Adelie body masses in one column, say A2:A152. Then =AVERAGE(A2:A152) returns 3,700.66, =STDEV.S(A2:A152) returns 458.57, and =COUNT(A2:A152) returns 151. In Google Sheets, STDEV gives the same sample standard deviation. =T.INV.2T(0.05, 150) returns the critical value 1.976. Then =CONFIDENCE.T(0.05, 458.57, 151) returns a margin of 73.74 g in one step; its inputs are the two-tail share, the sample standard deviation, and the sample size. Subtract the margin from the mean for the lower end and add it for the upper end.

Honesty note: the spreadsheet keeps every digit of t*, so its lower bound can read 3,626.93 g instead of 3,626.92 g. That one-hundredth of a gram is rounding, not a mistake. Both answers are right; say which way you rounded.

Misconception check — dividing by n instead of √n. Students often compute 458.57 ⁄ 151 ≈ 3.04 g and call it the standard error. It is tempting because "divide by the sample size" sounds natural. But the standard error divides by the square root of the sample size. With 151 birds, the slip makes the interval about 12 times too narrow. The cure: write SE = s ⁄ √n in full before you put numbers in, and ask whether the result is a sensible wobble for a mean of thousands of grams.

Memory hook: x̄, then SE, then margin, then two ends — never skip a step.

7. Interpretation: say what the interval means

The numbers are only half the job. The other half is one sentence that a reader can trust. The course uses one template for every confidence interval.

Worked example — the interpretation sentence. Fill in the template with the penguin interval.

The template. "We are [level] confident that the population mean [variable, in context] is between [lower] and [upper] [units]."
Filled in. We are 95% confident that the population mean body mass of Adelie penguins is between 3,626.92 g and 3,774.40 g.

Every part of that sentence does a job. "95% confident" names the method's track record. "Population mean" says the target is μ, not x̄ and not single birds. The two ends and the units say where μ plausibly sits.

The interval is about the mean, and only the mean. Look at how narrow it is compared with the birds themselves. The interval is 147.48 g wide, while single Adelie masses have s (the sample standard deviation) of 458.57 g. So many individual birds weigh far more or far less than any value in the interval. The interval describes μ, not the birds.

Misconception check — "95% of the penguins are in the interval". Students often read the interval as the range where most birds' weights fall. It is tempting, because the interval is written in grams, the same units as a single bird. But the interval estimates one number, the population mean. A typical bird sits about 458 g from the mean, far outside a window only about 147 g wide. The cure: always say "the population mean … is between", never "the penguins weigh between".

Pause and reflect — choose one. Pick either of the 2 prompts; they count the same.

(a) A news story says, "Average commute: 28 minutes, 95% confidence interval 26 to 30 minutes." A reader concludes that almost every commute takes 26 to 30 minutes. Explain the mix-up to that reader in two or three sentences.
(b) Write the interpretation sentence for a confidence interval about something you care about, such as hours of sleep, grocery spending, or a team's points per game. Use made-up numbers, and underline the words that keep it about the mean.

Memory hook: the interval is about the mean — not about single birds.

8. The margin of error: take it apart

The margin of error has three parts, and each one is a lever: the critical value (set by the confidence level), the spread s, and n (the sample size). Change one lever at a time and watch what happens. The margin of error is t* × s ⁄ √n.

Worked example — lever one, the sample size. What if only 10 birds had been weighed, with the same s?

New critical value. df = 9 and the critical value is 2.262.
New standard error. SE = 458.57 ⁄ √10 ≈ 145.01 g.
New margin. Multiply the critical value 2.262 by 145.01 g, and the margin of error is about 328.01 g.

Now compare: 328.01 g versus 73.74 g. That is a 4.4-fold larger margin for 10 birds versus 151 birds. Weighing 151 birds instead of 10 makes the margin of error about 4.4 times smaller. A field team deciding how many birds to weigh can see what each extra bird buys.

Try one more size. What if 25 birds had been weighed, with the same s? Then df = 24 and the critical value is 2.064. SE = 458.57 ⁄ 5 ≈ 91.71 g, because √25 = 5. Multiply the critical value 2.064 by 91.71 g, and the margin of error is about 189.29 g. So 25 birds land between 10 birds and 151 birds, as you would expect.

Notice why more birds help twice. The square root of n grows, so the standard error shrinks. And the degrees of freedom grow, so t* shrinks too.

Lever two, the confidence level. The critical values on the z* ladder are 1.645 for 90%, 1.960 for 95%, and 2.576 for 99%. A higher confidence level needs a larger critical value, so the interval gets wider. Asking to be more sure costs you precision. A 99% interval from the same data is wider than the 95% one, and a 90% interval is narrower.

Lever three, the spread. The margin of error is t* × s ⁄ √n, so it grows in step with s (the sample standard deviation). A sample with twice the standard deviation, at the same sample size and confidence level, has twice the margin of error. A bakery whose loaves vary a lot in weight needs a bigger sample than a bakery whose loaves barely vary, to pin down its mean equally well.

Honesty note: the margin of error covers only the random wobble of sampling. It does not cover a biased sample, a scale that reads 50 g heavy, or birds recorded in the wrong species. Those errors sit outside the margin, and a larger sample does not shrink them.

Misconception check — "double the sample, halve the margin". Students often think twice as many birds gives half the margin of error. It is tempting, because it sounds fair. But the standard error divides by the square root of n. Doubling n divides the standard error by only about 1.41. To halve it, you need about four times as many birds. The cure: whenever you change n, change √n, and let the square root do the scaling.

Memory hook: more sure means wider; more data means narrower.

9. The confidence level: what 95% does and does not mean

The phrase "95% confident" is the most misread sentence in this course. It sounds like a probability about one interval. It is not. It is a fact about the method.

The confidence level is the long-run share of samples for which the interval method captures the population parameter, such as 90%, 95%, or 99%; it describes the method, not the chance that one computed interval is right.

Go back to the die. You watched one interval capture 3.5 and another miss it. Now picture the same thing with penguins.

Worked example — many teams, many intervals. Imagine many field teams, each weighing its own 151 Adelie penguins.

Each team builds its own interval. Every team gets its own x̄ and its own s, so every team's interval sits in a slightly different place.
Count the captures. If many field teams each weighed their own 151 Adelie penguins and built a 95% t interval, about 95% of those intervals would capture μ and about 5% would miss it.
Look at one interval. For any one interval, μ is either inside or not; the 95% describes the method.

In words: about 1 team in 20 publishes an interval that misses the true mean, and no team can tell from its own data whether it is that team. That is why careful scientists repeat studies.

Once the interval 3,626.92 g to 3,774.40 g is computed, there is nothing random left in it. μ is a fixed number, and the two ends are fixed numbers. Either μ is in there or it is not. The 95% is your confidence in the method that produced it, earned by its track record across many samples.

Misconception check — "there is a 95% probability that μ is in this interval". Students often say this, and it is the most tempting wrong sentence in the course, because it sounds so close to right. But μ does not move around; after you compute the interval, it either contains μ or it does not. The 95% belongs to the procedure: it captures μ for about 95% of samples. The cure: use the template "we are 95% confident that the population mean … is between … and …", and if someone asks what "confident" means, describe the method, not this one interval.

Memory hook: trust the method's track record — never one interval's luck.

Key terms

point estimate — A single number computed from a sample and used as the best guess for a population parameter; the sample mean x̄ is the point estimate of the population mean μ. (first used: 1. Point estimates: one number and its wobble)
interval estimate — A range of plausible values for a population parameter, built as a point estimate plus or minus a margin of error, so that it shows both the guess and how far off the guess might be. (first used: 2. Interval estimates: turn the wobble into a range)
margin of error — The distance from the point estimate to either end of a confidence interval, equal to the critical value times the standard error; it measures sampling variability only, not bias. (first used: 2. Interval estimates: turn the wobble into a range)
confidence interval — An interval estimate built by a method that captures the population parameter in a stated share of all possible samples, such as 95%; it is reported as an estimate plus or minus a margin of error. (first used: 2. Interval estimates: turn the wobble into a range)
t distribution — A family of symmetric, bell-shaped curves centered at 0 with heavier tails than the standard normal distribution, used for a mean when the sample standard deviation s stands in for σ; it grows closer to the standard normal as the degrees of freedom grow. (first used: 3. The t distribution: pay for an unknown spread)
degrees of freedom — The number, written df, that picks which t distribution to use; for one sample of size n it is the sample size minus 1. (first used: 3. The t distribution: pay for an unknown spread)
critical value — The number of standard errors a margin of error reaches for a chosen confidence level, read from the standard normal distribution (z*) or a t distribution (t*); for 95%, z* is 1.960. (first used: 4. The critical value: read it off the ladder for your sample)
one-sample t interval — The confidence interval for a population mean when σ is unknown, x̄ plus or minus t* times s divided by the square root of n, used after checking for a random or representative sample, independent observations, and a roughly normal population or a large enough sample. (first used: 6. The one-sample t interval: build it by hand and by spreadsheet)
confidence level — The long-run share of samples for which the interval method captures the population parameter, such as 90%, 95%, or 99%; it describes the method, not the chance that one computed interval is right. (first used: 9. The confidence level: what 95% does and does not mean)

Summary

A point estimate is one number; an interval estimate adds a margin of error so the reader can see the wobble.
About 95% of sample means land within about two standard errors of the true mean, so an interval that reaches that far around x̄ captures μ for about 95% of samples.
When σ is unknown, use s in its place and a t distribution with the sample size minus 1 degrees of freedom.
The critical value t* is larger for small samples and shrinks toward z* as the sample grows.
Check the three conditions first: a random or representative sample, independence, and a roughly normal population or a large sample.
The one-sample t interval is the sample mean plus or minus t* times the standard error.
Interpret every interval as "we are 95% confident that the population mean … is between … and …".
A higher confidence level or more spread widens the interval; a larger sample narrows it, and halving the margin takes about four times the data.
The 95% describes the method's long-run capture rate, not one interval and not the share of individuals inside it.

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

A coffee roaster weighs a random sample of 25 bags from one day's packing. The bags average 342 g with a sample standard deviation of 5 g. Find the 95% confidence interval for the mean weight of all bags packed that day, and write the interpretation sentence.
A sleep study measures 10 volunteers. For a 95% interval, what are the degrees of freedom and the critical value t*? About how much farther does that t* reach than the normal critical value 1.960?
A city survey reports, "We are 95% confident that the mean one-way commute of city workers is between 24 and 30 minutes." Your friend says, "So there's a 95% chance the true mean is between 24 and 30." What would you tell your friend?
A school nurse builds a 95% interval for students' mean resting heart rate from 40 students and gets a margin of error of 4 beats per minute. About how many students would the nurse need for a margin of about 2 beats per minute, with the same spread? Explain why.

Answers (worked reasoning — read after attempting)

The sample size is 25, so df = 25 − 1 = 24 and the critical value t* for 95% is 2.064. The standard error is 5 ⁄ √25 = 5 ⁄ 5 = 1 g. Multiply the critical value 2.064 by 1 g, and the margin of error is about 2.06 g, rounded to two decimals. The interval runs from 342 − 2.06 = 339.94 g to 342 + 2.06 = 344.06 g. We are 95% confident that the population mean weight of bags packed that day is between 339.94 g and 344.06 g. The roaster can decide the 340 g label is believable for the average bag, though single bags can still come in light.
The degrees of freedom are 10 − 1 = 9, and the critical value t* for 95% with 9 degrees of freedom is 2.262. Compare it with 1.960: 2.262 ⁄ 1.960 ≈ 1.15, so the t interval reaches about 15% farther. That extra reach pays for guessing σ from only 10 people.
Tell your friend that the true mean is a fixed number, so once the interval is computed it either contains the mean or it does not. The 95% describes the survey's method: about 95% of intervals built this way capture the true mean. So "95% confident" is about the track record of the method, not a 95% chance for this one interval.
About 160 students, four times as many. The margin of error divides by the square root of the sample size, so halving it means doubling √n, and doubling √n means multiplying n by 4: 4 × 40 = 160. The critical value also shrinks a little with more students, so 160 is a slight overestimate, which is the safe direction for planning.

Next up in this module: the Readings & Resources page, the lecture tutorial, the practice set, and the quiz — and next week the same margin-of-error idea builds a confidence interval for a proportion.

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