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Week 11 · Chapter

Chapter 11 — Confidence Intervals for a Mean

Introduction to Statistics Generic evergreen edition

Course: Introduction to Statistics (18-week generic edition)
The module's primary reading · ~30–40 minutes · Objective 6 · Read this before (or alongside) the week's lecture — everything else in the module builds on it. The friendly t-table lives in Section 2.

Why you already care about this

Your sleep app announces: you average 6.9 hours a night. One number, two decimal places, total confidence. A laptop review declares battery life: 10.2 hours. A café's sign promises a 355 mL pour. Here is the question almost nobody asks those numbers: how wrong could you be? Every one of them came from a sample — some nights, some laptops, some cups — and Week 10 taught you exactly what sample answers do: they wobble. A single number pretends the wobble away. It is, politely, a bluff.

This week you learn the honest alternative. Instead of one number that is almost surely not exactly right, you'll report a range you can defend — a confidence interval — and attach a truth-in-labeling statement to it: how confident you are, and in precisely what sense. By the end of this chapter you will be able to: name the point estimate for a mean and say why it needs a cushion; explain why a new curve called t replaces the normal's z when the population's spread is unknown; construct a one-sample t-interval from any small dataset (conditions, standard error, margin of error, endpoints); take a margin of error apart and predict what makes intervals wider or narrower; and interpret the finished interval without committing either of the two classic misreads.

Carry this line through the week: an honest estimate is a range with a confidence attached — one number is a bluff.

Where we are

The course's first half taught you to describe data you have. Week 10 opened the second half with the engine of inference: take repeated samples and the sample mean x̄ dances around the true mean μ in a predictable way — centered on the truth, with a typical miss called the standard error. One sentence of review, because everything below rides on it: individuals spread with standard deviation s, but a sample mean is steadier, wobbling by only SE = s/√n. This week that engine finally drives somewhere: from "my sample's answer wobbles" to "here is the range of truths my sample can defend."

1. Point estimates, and the interval idea

Start with the plain idea. When you want a population mean μ — the true average sleep of all members, the true average fill of all cups — your sample hands you its best single guess: the sample mean x̄. That single best guess is called a point estimate. (Week 1's letters still line up: μ is the parameter we want; x̄ is the statistic we have.)

The point estimate's problem is not that it's a bad guess. It's that, as a single point, it is almost surely not exactly μ. Different random samples give different x̄'s — that's sampling variability, the lesson you watched happen in Data Lab 1 and formalized in Week 10. Expecting one sample's mean to land exactly on the truth is like expecting a dart to land on a mathematical point.

The fix is old, simple, and honest: report the guess plus or minus some wiggle room.

A confidence interval (CI) for a mean is: point estimate ± margin of error — the sample's best guess, widened by a cushion built from how much guesses like it typically miss.

The margin of error (ME) has exactly two ingredients, and you already own one of them:

  • How much guesses like ours wobble: the standard error, SE = s/√n — Week 10's engine.
  • How many wobbles of cushion we want: a multiplier chosen by the confidence level — 90%, 95%, or 99%. More confidence, bigger multiplier, wider interval.

So the whole recipe will be center ± (multiplier × wobble). The only missing piece is the multiplier — and it hides a small surprise.

Misconception check. "My sample mean didn't equal the true mean, so my sample failed." No — an estimate is not a prophecy. Missing by a predictable amount is the method working; the entire construction below runs on knowing the typical miss. The only real failure is pretending there is no miss at all.

2. Why not z? The t-distribution and the friendly table

Week 8's normal machinery ran on z, and z has a hidden requirement: it assumes you know the population standard deviation σ. In real life, nobody hands you σ. You estimate it with the sample's own s — and s, being computed from the sample, wobbles too. If you use the z multiplier anyway, you ignore that second layer of wobble and act more certain than you are.

The honest fix is a slightly different curve: the t-distribution. It looks like the normal — bell-shaped, symmetric, centered at 0 — but with heavier tails: values far from center are a bit more likely. The heavy tails are the price of estimating σ. A memory hook that carries the whole idea: "t is z with humility."

There is one more wrinkle: there isn't a single t curve, but a family, one for each sample size, indexed by degrees of freedom:

df = n − 1 — one degree of freedom is spent estimating the mean before s can be computed.

Small samples → heavier tails → bigger multipliers. As n grows, s pins down σ better and better, and the t curve melts into the normal.

The multiplier itself is called the critical value, t*: the number of standard errors you must reach out from the center to enclose 90%, 95%, or 99% of the curve. This course runs on a friendly t-table — the same table appears in the tutorial and the assignment coach, and every problem this week lands exactly on it.

The table below gives the two-sided critical value t* for this week's three sample sizes, plus the z* row that t approaches.

df (sample size) 90% 95% 99%
9 (n = 10) 1.833 2.262 3.250
15 (n = 16) 1.753 2.131 2.947
24 (n = 25) 1.711 2.064 2.797
z* (very large n) 1.645 1.960 2.576

Read the table three ways, out loud if it helps:

  • Down a column: at 95%, t* falls 2.262 → 2.131 → 2.064 → 1.960. More data means less humility is needed; t* slides toward z*.
  • Across a row: at df 24, the multiplier climbs 1.711 → 2.064 → 2.797. More confidence costs a bigger cushion.
  • The corner: df 9 at 99% is 3.250, the largest value on the table. Little data plus a big promise requires the widest cushion of all — as it should.

Worked example. A reviewer will test n = 10 laptops: df = 10 − 1 = 9, so a 95% interval uses t* = 2.262. A barista will sample n = 16 pours: df = 15, and 95% gives t* = 2.131. Both are larger than z* = 1.960 — every t interval is a little wider than the z version would be. That extra width is not waste; it's honesty about having estimated σ.

Misconception check. "n = 16, so I look up row 16." The row is df = n − 1 = 15. Say the subtraction out loud every time until it's a reflex — the n-vs-(n − 1) slip is this week's most common table error. And its sibling: "1.96 is close enough." At n = 10 the honest multiplier is 2.262; sneaking in 1.96 shaves about 13% off your margin and quietly overstates your certainty. (It is also the single most common chatbot error this week — Section 5's tools passage shows you how to catch it.)

3. Building the one-sample t-interval

Everything is now on the bench: a center (x̄), a wobble (SE = s/√n), and a multiplier (t* from the table). Before assembling, check the password.

The conditions:

  1. Random. The data are a random sample (or something defensibly like one) from the population you're describing. Week 1 never retires: no formula can launder a biased sample.
  2. Roughly normal — it matters when n is small. For samples of about 25 or fewer, the data should show no strong skew and no wild outliers — glance at a dot plot or histogram before trusting the machinery. For larger samples, Week 10's Central Limit Theorem carries the load.

The recipe (four beats, in order):

① Check the conditions → ② SE = s/√n → ③ ME = t* × SE (using df = n − 1 and your confidence column) → ④ Interval = x̄ ± ME. Then say the interval in words.

Worked example — the week's centerpiece. A wellness program records nightly sleep for a random sample of n = 25 members: x̄ = 7.2 hours, s = 1.0 hour. Build a 95% confidence interval for the mean nightly sleep of all members.

  1. Conditions: random sample given; n = 25 with no strong skew reported. Pass.
  2. SE = 1.0/√25 = 1.0/5 = 0.2 hours. (A 25-person mean is five times steadier than one person.)
  3. df = 24; the 95% column gives t* = 2.064; so ME = 2.064 × 0.2 = 0.4128 ≈ 0.41 hours.
  4. Interval: 7.2 − 0.41 to 7.2 + 0.41 → (6.79, 7.61) hours — unrounded, 6.7872 to 7.6128; round at the end, not in the middle.

In words: "We are 95% confident that the mean nightly sleep of all members is between about 6.8 and 7.6 hours." Notice what the sentence is about: the mean of the population — not any individual sleeper.

Worked example — smaller sample, different row. A café's espresso machine pours a "355 mL" house coffee. A barista randomly pulls n = 16 cups: x̄ = 354 mL, s = 8 mL. For a 95% interval: SE = 8/√16 = 8/4 = 2 mL; df = 15 so t* = 2.131; ME = 2.131 × 2 = 4.262 mL; interval 354 ± 4.262 → (349.7, 358.3) mL.

Now read it like an analyst: the label's 355 sits inside the interval. A true mean fill of 355 is entirely plausible — this sample builds no case against the machine. A claimed mean of 360, by contrast, sits outside: the data push back. This "menu of plausible values" reading is the second correct way to use any interval — and it is Week 13's hypothesis test wearing pajamas.

In practice (your tools). The whole construction is four cells in a spreadsheet. With your sample in A2:A26: =AVERAGE(A2:A26) gives x̄, =STDEV(A2:A26) gives s, =STDEV(A2:A26)/SQRT(25) gives the SE. For the multiplier, =T.INV(0.975, 24) returns 2.0639 — the same 2.064 as the friendly table (0.975 because 2.5% sits in each tail; df = 24; Google Sheets and Excel are identical here). There is even a one-liner for the margin: =CONFIDENCE.T(0.05, 1.0, 25) returns 0.4128 for the sleep data. Two sanity habits: the finished interval should have x̄ dead-center, and ME should be bigger than SE — if either fails, hunt the typo.

Misconception check. Rounding early. If you round SE or t*·SE in the middle of the work, your endpoints stop reconciling with each other. Carry full digits to the final step; round only the endpoints, and say how much you rounded ("≈ 0.41").

4. Margin-of-error anatomy: what moves the width

Write the margin out in full — ME = t* × s/√n — and you can see every dial that controls an interval's width:

  • Confidence level ↑ → wider. The only way to promise more certainty from the same data is a bigger t*, which buys a bigger cushion.
  • Sample size ↑ → narrower. More data shrinks SE — but through √n, so quadrupling the sample only halves the margin. Precision is bought at a steep exchange rate; this is why careful studies are expensive, and why a suspiciously tight interval from a tiny sample deserves a raised eyebrow.
  • Spread s ↑ → wider. Noisier populations produce wobblier means. You don't choose s; you inherit it from reality.

Worked example — one dataset, three honest reports. The sleep data again (x̄ = 7.2, SE = 0.2, df = 24), at three confidence levels:

  • 90%: ME = 1.711 × 0.2 = 0.3422 → interval (6.86, 7.54) — narrowest, least sure.
  • 95%: ME = 2.064 × 0.2 = 0.4128 → interval (6.79, 7.61) — the everyday convention.
  • 99%: ME = 2.797 × 0.2 = 0.5594 → interval (6.64, 7.76) — surest, widest.

Nothing about the data changed — only how much certainty was purchased. Certainty costs width. (Push it to absurdity once, so it sticks: a 99.999% interval would be nearly certain and nearly useless — "your mean sleep is between 2 and 12 hours.")

Reverse reading (you'll use this in the assignment). Any reported interval can be taken apart with two subtractions: center = (low + high)/2 = x̄, and ME = (high − low)/2. An interval of (83.7, 100.3) seconds was centered at x̄ = 92 with a margin of about 8.3. No one can hide the anatomy from you.

Misconception check. "To halve the margin, double the sample." The √n is in the basement of the formula: halving ME takes four times the sample (√4 = 2). Doubling n only shrinks the margin by a factor of √2 ≈ 1.4.

5. Interpretation: what "95% confident" actually buys

The interval is built. Now the harder skill — saying what it means without lying. Take the third worked example of the week:

Worked example. A supermarket times a random sample of n = 25 self-checkout transactions: x̄ = 92 seconds, s = 20 seconds. SE = 20/√25 = 4; df = 24, t* = 2.064; ME = 2.064 × 4 = 8.256; the 95% interval is (83.7, 100.3) seconds.

The correct sentence: "We are 95% confident that the mean self-checkout time for all transactions is between about 84 and 100 seconds."

What the "95%" describes: the method, not this one interval. Imagine many teams, each drawing its own random 25 transactions and building its own interval this way. About 95% of those intervals would capture the true μ — and about 1 in 20 would honestly, blamelessly miss. Your interval is one ticket from that batch: probably a winner, possibly not, and you cannot tell which from the inside — μ never announces itself. That long-run capture rate is the confidence level's entire meaning.

Now put the two classic misreads on trial:

  • ❌ "95% of customers finish between 83.7 and 100.3 seconds." Guilty — of confusing individuals with the mean. The interval hunts μ; individual customers spread with s = 20, so plenty finish in 60 seconds and plenty take 130. Most individuals live outside a good interval for the mean, and that's healthy: means are steadier than people.
  • ❌ "There's a 95% probability that μ is in (83.7, 100.3)." Gently guilty. μ is a fixed number, not a spinner — it is in this particular interval or it isn't; nothing is left to chance once the interval exists. The chance lived in the sampling: 19 of 20 such intervals capture μ. The licensed shortcut for all of this is exactly the phrase "we are 95% confident." Use it, and know what it stands for.

And keep the second reading from Section 3 in your pocket: an interval is a menu of plausible values for μ. "Could the true mean checkout time be 100 seconds?" — 100 is (barely) inside: plausible. "Could it be 110?" — outside: the data push back. Next-next week, that push gets a name, a formula, and a verdict.

In practice (your tools). This is also the week to catch your chatbot in the act. Ask it to build a 95% interval for a small sample (say, 10 laptops with x̄ = 9.5 hours, s = 1.5) and check the multiplier it used against the friendly table. The honest value is t* = 2.262 (df 9). Chatbots routinely grab z* = 1.96 — producing an interval about 13% too narrow that quietly overstates certainty — or quote a t-table "from memory" and drift by a digit. The tool drafts; you judge. This exact catch is graded in Data Lab 11.

Misconception check. "My interval contains the true mean — I checked, it looks right." Nobody ever knows whether their interval captured μ. That un-knowability is precisely why the confidence statement exists. This week's data lab makes the idea visible: every student builds an interval from their own random penguin sample around a knowable full-data mean — and roughly 95% of the room captures it, while a few honest intervals miss. The misses aren't errors. They're the 1 in 20, showing up on schedule.

Key terms

  • Point estimate — a sample's single best guess at a parameter; for a mean, the sample mean x̄.
  • Confidence interval (CI) — point estimate ± margin of error; a range of values for the parameter, with a stated confidence level.
  • Confidence level — the method's long-run capture rate: the percentage of such intervals (over many samples) that would contain the true parameter.
  • Margin of error (ME) — the interval's half-width: ME = t* × SE.
  • Standard error (SE) — the typical wobble of the sample mean: SE = s/√n.
  • t-distribution — the bell-shaped curve with heavier tails used when σ is estimated by s; one curve per degrees of freedom.
  • Degrees of freedom (df) — the t-family index: df = n − 1 for a one-sample interval.
  • Critical value (t*) — the multiplier from the t-table that sets how many SEs of cushion the confidence level requires.
  • One-sample t-interval — x̄ ± t*·(s/√n), valid when the sample is random and, for small n, the data are roughly normal.
  • Conditions — the password before the formula: a random sample; no strong skew or wild outliers when n is small.
  • Capture rate — the fraction of intervals, over repeated sampling, that contain the true parameter — what the confidence level promises.
  • Plausible values — the values of μ inside the interval: claims the data can live with; values outside, the data push back on.

Summary

  • A point estimate (x̄) is the best single guess and is almost surely not exactly μ — the honest report is x̄ ± margin of error.
  • With σ unknown, s stands in — and the multiplier comes from the t-distribution with df = n − 1: heavier tails, bigger multipliers for smaller samples, melting into z as n grows. t is z with humility.
  • The recipe: conditions (random; roughly normal for small n) → SE = s/√n → ME = t* × SE → x̄ ± ME — then say it in words, about the mean.
  • Width has three dials: confidence level ↑ widens, n ↑ narrows (by √n — halving the margin costs 4× the data), s ↑ widens. Certainty costs width.
  • "95% confident" describes the method: about 19 of 20 such intervals capture μ; yours is one unlabeled ticket. It is not "95% of individuals" and not "a 95% probability for this fixed interval."
  • Bonus reading: the interval is the menu of plausible values for μ — inside is plausible, outside the data push back. That's Week 13 warming up.

Check your understanding

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

  1. A hospital scheduling office records the nightly sleep of a random sample of 16 rotating-shift nurses: x̄ = 6.3 hours, s = 0.8 hours. Build the 95% confidence interval (use the friendly table), and write the one-sentence interpretation.
  2. Explain in your own words: why is t* always bigger than z* at the same confidence level — and what happens to t* as the sample grows? (Use the 95% column: 2.262 → 2.131 → 2.064 → 1.960.)
  3. A tech site reports a 99% confidence interval of (7.61, 8.39) hours for the mean video-playback battery life of a gaming laptop model. From the endpoints alone, recover the sample mean and the margin of error.
  4. A coffee truck posts a 95% confidence interval of (452, 468) mL for the mean fill of its large americano. A regular says: "So 95% of larges are between 452 and 468 mL — and there's a 95% chance the true mean is in there." Identify both misreads and write the sentence the regular should say.

Answers (worked reasoning — read after attempting)

  1. Conditions: random sample stated; n = 16 is small, so we'd also want no wild skew — proceed. SE = 0.8/√16 = 0.8/4 = 0.2 hours. df = 16 − 1 = 15, so t* = 2.131. ME = 2.131 × 0.2 = 0.4262 ≈ 0.43. Interval: 6.3 ± 0.43 → (5.87, 6.73) hours. Sentence: "We are 95% confident that the mean nightly sleep of all rotating-shift nurses at this hospital is between about 5.9 and 6.7 hours." (About the mean — not any one nurse.)
  2. t* is bigger because the interval must budget for two wobbles: the sample mean's wobble and the wobble in s, which stands in for the unknown σ. The heavier tails of the t curve are that budget. As n grows, s estimates σ more reliably, the extra wobble shrinks, and t* melts toward z* — exactly what the 95% column shows: 2.262 (n = 10) → 2.131 (n = 16) → 2.064 (n = 25) → 1.960 (huge n). t is z with humility, and humility fades with evidence.
  3. Center = (7.61 + 8.39)/2 = 8.00 hours — that's x̄. Margin = (8.39 − 7.61)/2 = 0.39 hours. Any interval surrenders its anatomy to two subtractions: the guess was 8.00, the cushion 0.39.
  4. Misread 1: individuals vs. the mean — the interval describes the mean fill of all larges, not 95% of cups; individual pours spread more widely than means do. Misread 2: probability vs. confidence — the true mean is a fixed number, in the interval or not; the 95% describes the method's capture rate (about 19 of 20 such intervals catch the truth). The sentence to say: "We are 95% confident that the mean fill of all large americanos is between 452 and 468 mL — the range was built by a method that captures the true mean about 19 times in 20."

Next up in this module: the slide deck if you want the lecture's visual walk-through, then the Readings & Resources page for optional deeper dives — and when you're ready to practice, Lecture Tutorial 11 turns your chatbot into your personal Week 11 tutor (friendly t-table included).