L-INF-2: Confidence Intervals for a Mean

Module 4 · Statistical Inference

Section 1: Introduction

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

You and a sceptical colleague are about to run the same careful study, separately: each of you will time 50 random coffee orders at a café, from order to pickup. You will report a 95% confidence interval for the café’s mean wait; your colleague will report their sample mean.

Before either of you collects any data, make a bet. Imagine running this pair of studies over and over. How often will your colleague’s sample mean land inside your interval: more than 95% of the time, about 95%, somewhere between 75% and 95%, or less than 75%?

Pause and write. Write your bet and one sentence of reasoning on paper or in your notes. You will settle it in the Boss Fight.

Most people answer “about 95%”. Whether that is right depends on what the 95% actually promises, and that is what this lesson is about.

In L-INF-1 you knew the machine’s true mean and asked where sample means land. Real studies run the other way. You have one sample and one , and the true mean is hidden. A confidence interval turns that one sample into an honest range for , together with a stated track record for the method that produced it.

After this lesson, you will be able to:

Construct a confidence interval for using , and check when this z procedure applies
Choose the appropriate for 90%, 95%, and 99% confidence
Interpret an interval correctly, and use it to decide whether a population mean is above or below a threshold
Predict how confidence level, sample size, and population variability change interval width
Determine the minimum sample size needed for a target margin of error using

This lesson’s z procedure assumes that the population standard deviation is known. L-INF-3 handles the common case where is unknown by using the t-distribution.

Section 2: Prerequisites

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Confidence intervals are a direct application of the sampling distributions you studied in L-INF-1.

From L-INF-1 — Standard error: measures the typical distance between and ; today it becomes the yardstick for the interval.
From L-INF-1 — Where sample means land: Example 5 found that the middle 95% of sample means fall within of . Today you run that idea backwards.
From L-INF-1 — Sampling distributions and the CLT: an independent random sample from a population that is normal, or a sample large enough for to be reasonably normal despite the population’s shape, supports the normal model used here. The familiar guideline is a screening heuristic, not a guarantee.
From L-PR-6 — Normal areas and critical values: a central confidence level leaves equal tail areas; today you translate those areas into .
From L-DS-1 — Random sampling: probability-based inference needs a sampling process that represents the target population; every interval conclusion inherits that design condition.

Retrieval Checkpoint

If a sampling distribution is normal and we want to capture the middle 95% of all possible sample means, how many standard errors (SE) away from the mean must we go in both directions?

I can explain why averages vary less than individual observations. I can compute

without replacing

. I can translate a central normal area into two equal tail areas.

Bridge to today: L-INF-1 told you how sample means vary around a known . In this lesson the direction reverses: one observed is all you have, and you ask which values of it is consistent with. The CLT and standard error carry forward; the interval and its interpretation are new.

Section 3: Core Concepts

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Your six-part interval workflow:

C1 — Estimate: move from one best guess to a range.
C2 — Build: find the values of your is consistent with: .
C3 — Set confidence: translate the confidence level into .
C4 — Interpret: the confidence level is the method’s track record; use the interval to decide.
C5 — Compare precision: track how , , and change width.
C6 — Plan: solve backward for the minimum sample size.

C1 — From Point Estimate to Interval Estimate

When you compute from a sample, you get a point estimate of . It’s your single best guess. But it’s almost certainly not exactly right: a different sample would give a different . A point estimate alone gives you no information about how wrong you might be.

An interval estimate (confidence interval) attaches a range to your estimate: . The “something” is the margin of error. It quantifies the precision of your estimate.

Margin of Error

The margin of error is the half-width of the confidence interval. It equals the critical value times the standard error:

A smaller margin of error means a more precise (narrower) interval. It decreases when you increase or accept a lower confidence level.

A sample reports and . The point estimate is 72; the interval estimate is . The range shows the uncertainty that the point estimate alone hides.

A confidence interval does not claim that every value inside the range is equally plausible, nor that observations themselves must fall inside it. It is an estimate of one population mean.

C2 — Which Settings Fit Your Sample?

Go back to L-INF-1’s cereal machine: g, and inspectors average boxes, so g. Example 5 found that, whatever the machine’s setting , the middle 95% of sample means fall within g of that setting.

Now the setting is hidden. An inspector’s sample of 36 boxes averages g. Which settings could have produced it? Call a setting plausible if 498.6 g falls inside that setting’s middle 95%. Before reading on, try at least four settings in Figure 1 and watch which ones fit.

Candidate setting μ = 505 g

Setting 505 g: its middle 95% of sample means runs 501.08 to 508.92 g, so 498.6 g would be unusual. Does not fit.

Figure 1: Move the candidate setting. Its orange band shows where the middle 95% of sample means land for that setting; the green line is your sample mean. Each setting you try leaves a mark. When you have a feel for the pattern, reveal every setting that fits.

The settings that fit run from g to g. That is no coincidence. Your is within 3.92 g of a setting exactly when that setting is within 3.92 g of your : distance is the same in both directions. So the plausible settings are , and that range is the 95% confidence interval for .

Confidence Interval for μ (Large Sample, σ Known)

which gives the interval:

Conditions required:

Independent random sample; when sampling without replacement, sample no more than 10% of the population or use the finite-population correction
The population is approximately normal, or the sample is large enough that extreme skew, heavy tails, and influential outliers do not undermine the normal approximation for
is known; if it is unknown, use the t procedure developed in L-INF-3

The three ingredients you need are: (1) your sample mean , (2) the standard error , and (3) the critical value that matches your desired confidence level. When a condition fails, no amount of careful arithmetic rescues the interval: a non-random sample has no track record to quote.

Do not use as the standard error of a mean. Individual observations vary by ; sample means vary by . Skipping the square-root adjustment makes the interval far too wide.

C3 — Critical Values for Common Confidence Levels

The critical value is the z-score that captures the middle C% of the standard normal distribution. For a 95% CI, you want to leave 2.5% in each tail, so is the value with left-tail area 0.975.

Confidence Level	Tail area each side	Rounded critical value
90%	0.05	1.645
95%	0.025	1.96
99%	0.005	2.576

These three rounded values are worth memorizing; they appear in every inference problem through this module. They are not magic constants: each is derived from the standard normal table you studied in L-PR-6. For a 95% CI, the middle 95% leaves 2.5% in each tail, so you need the z-score where ; an inverse-normal calculation gives . Challenge 2 in Section 9 asks you to build a critical value for an unfamiliar confidence level and use it in a complete interval.

Figure 2: The anatomy of z*. The grey region contains the middle C% of the standard normal distribution; the orange tails together hold the remaining (1−C)%. Outward arrows show that the tails continue beyond the finite plotting window. The dashed lines mark ±z* — a distance along the z-axis, not an area. Toggle confidence levels to see how z* is derived from the inverse cumulative normal. At 95%, the faint dashed line at 1.645 is where Φ(z) = 0.95 lands: that is the 90% value, because 0.95 is the area to the left of z*, not the middle area.

Students sometimes confuse confidence level with z-score direction. For a 95% CI: the middle 95% of the standard normal spans from to . The critical value is 1.96, not 0.95 and not the z-score for 0.95 (which would be 1.645, the 90% critical value). Always look up z* as the value where left-tail area = , where (confidence level).

C4 — What the 95% Promises

Before you open Figure 3, make a prediction. If 100 different inspectors each build a 95% interval from their own random sample, about how many of the 100 intervals will miss the true mean? Write a number down.

Sample size n: 50

Confidence level

Show μ to score the intervals.

Did your interval capture μ? With μ hidden, there is no way to tell.

Figure 3: Each bar is a confidence interval x̄ ± z*·σ/√n built from one random sample (σ = 15 known). The green bar is your interval; the thin orange bars come from other samples drawn the same way. μ starts hidden: from one interval alone you cannot tell whether it caught μ. Show μ, then draw 100 more and count the misses. Changing the confidence level re-scores the same samples with new widths.

With hidden, there is no way to know whether your own interval caught it. What you can know is the record of the method: across many samples, about 95 of every 100 intervals built this way capture , and about 5 miss. That record is what “95% confidence” means.

What a Confidence Interval Means

A 95% confidence interval does not mean: “There is a 95% probability that lies in this interval.”

The true population mean is a fixed (though unknown) constant. Once you’ve computed your interval, either is or isn’t inside it.

The correct statement is: “If we repeated this procedure many times, each time drawing a new sample and computing a new interval, about 95% of those intervals would contain .”

The 95% refers to the process, not to any single interval.

Two accepted ways to say it. In textbooks, reports, and the rest of this course you will read: “We are 95% confident that the true mean lies between [L] and [U].” This is shorthand for the procedure statement above, not a probability about the fixed . A second, very useful reading comes straight from C2: the interval is the range of plausible values for , the values your sample is consistent with. That reading is how you make decisions with an interval: if a threshold lies outside the interval, the data settle which side is on; if it lies inside, they do not.

The interval is narrow because averages are steady. Individual values are not, which is why the interval says almost nothing about where single observations fall.

Only 12 of 50 waits fall inside the interval for the mean. Sample 1: x̄ = 101.86 s; 95% CI for the mean ≈ 97.70 to 106.02 s.

Figure 4: One sample of 50 café waits (green dots) and its 95% interval for the mean (green bar). Count how many individual waits fall inside the interval, then draw a new sample. The interval estimates the mean; it is not a range for customers.

Three readings that sound right and are wrong:

“There is a 95% probability that is in this interval.” The probability belongs to the method; is fixed.
”95% of individuals fall in this interval.” The interval is about the mean; individual values spread far wider.
”A repeat study’s mean will land in this interval 95% of the time.” Both studies wobble, so it happens less often. You will work out how often in the Boss Fight.

C5 — What Affects the Width of a Confidence Interval?

The width of a CI is . Three factors determine it:

Confidence level ↑ → width ↑: A higher confidence level requires a larger (e.g., 2.576 vs. 1.96). More confidence = wider interval.
Sample size ↑ → width ↓: Larger shrinks the SE = . To halve the width, you must quadruple .
Population variability ↑ → width ↑: Larger means individual observations are more spread out, so sample means are less precise.

Confidence level

Same data: the 99% interval is always 56.6% wider than the 90% one (2.576 ÷ 1.645).

Sample size n = 50

Population SD σ = 15

Change one knob: the old interval stays above as “before”.

Standard error: SE = σ/√n = 15/√50 ≈ 2.12
Critical value: z* = 1.96 (95%)
Margin of error: E = z* × SE ≈ 4.16
Interval: Interval: 100 ± 4.16 ≈ (95.84, 104.16)

Values are rounded to two decimals; the bounds and the × ratios use the unrounded E.

Figure 5: One sample, one interval (green, centred on x̄). Change one knob at a time and compare the new interval with the dashed “before” outline. More confidence or a larger σ widens it; a larger n narrows it, but only by √n: four times the sample halves the margin of error.

Study design can control and select a defensible confidence level. The population’s is not changed by the study. Lowering confidence narrows the interval, but it also raises the long-run miss rate, so confidence should be chosen for the decision rather than adjusted only to make the result look precise.

Doubling does not halve the margin of error. Because contains , halving requires about four times the sample size when and stay fixed.

C6 — Planning Ahead: Determining Required Sample Size

Before collecting data, you often want to guarantee a maximum margin of error . You can solve for directly:

Minimum Sample Size for a Desired Margin of Error

Set and solve for :

The ceiling function () means always round up to the next whole number; rounding down would give you a margin of error slightly larger than desired.

Required inputs: desired confidence level (gives ), known or estimated , and target margin of error .

Figure 6: The cost of precision. Drag the dark dot, click anywhere on the chart, or focus it and use the arrow keys to set your desired margin of error E. The purple diamond shows what happens if you halve E — the required sample size approximately quadruples (half the error means roughly 4× the sample), because n scales as 1/E². Notice how the curve accelerates steeply for small E: asking for twice the precision is far more expensive than it looks.

Always round the sample size up, never down. If the formula gives , the answer is 62, not 61. With your margin of error would exceed the target; you need that extra observation to meet the specification.

Section 4: Worked Examples

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Example 1 — Constructing a 95% CI (Fully Worked)

A coffee shop times the wait from order to pickup. Years of records show no extreme skew or influential outliers and give minutes. An independent random sample of orders yields minutes. Construct a 95% confidence interval for the true mean wait time .

How I choose the method: I notice that the target is a population mean, the sample is random and large, and is known. That combination points to a large-sample z interval. I will use the sample mean as the centre and as the uncertainty yardstick.

Step 1 — Verify conditions:

Independent random sample: ✓ (stated)
with no extreme skew or influential outliers: ✓ normal approximation for is reasonable
is known: ✓

Step 2 — Identify the critical value:

Step 3 — Compute the standard error and margin of error:

Step 4 — Build the interval:

Interpretation: We are 95% confident that the true mean wait time is between 3.90 and 4.50 minutes. This means we used a procedure that captures the true mean in about 95% of samples drawn this way. Equivalently, mean waits between 3.90 and 4.50 minutes are the plausible values: the ones this sample is consistent with.

Example 2 — Constructing a 90% Known-σ Interval and Making a Decision (Partially Scaffolded)

A health researcher measures resting systolic blood pressure in an independent random sample of adults. Assume the population distribution has no extreme skew or influential outliers. Long-term validated records give mmHg, and the sample yields mmHg. Construct a 90% confidence interval. A clinic also asks whether this population’s mean exceeds 125 mmHg: say what the interval allows you to conclude.

Method boundary: The population SD mmHg is known, so this lesson’s z procedure applies. If only the sample SD were available, the t procedure from L-INF-3 would be the appropriate method.

Before opening the solution, identify the critical value for 90% confidence, compute the standard error from the known , construct the interval, interpret it in context, and answer the clinic’s question.

This first attempt matters: committing an answer, even an uncertain one, makes comparison with the model more useful.

Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.

Show Solution

Step 1 — Conditions: independent random sample, no extreme skew or influential outliers, and known mmHg; the z interval is appropriate.

Step 2 — Critical value:

Step 3 — SE and margin of error:

Step 4 — Interval:

Interpretation: We are 90% confident the true mean resting blood pressure of this population is between 124.1 and 131.9 mmHg.

The clinic’s question: 125 mmHg lies inside the interval. Means a little below 125 are still plausible, so these data cannot settle whether the population mean exceeds 125 mmHg, even though the sample mean (128) does. A larger sample would narrow the interval and might settle it.

Example 3 — Comparing Widths Across Confidence Levels (Minimally Scaffolded)

A museum takes an independent random sample of visitor sessions from a visit-duration distribution with no extreme skew or influential outliers. The mean visit duration is minutes, and historical records give a population SD of minutes. Compute and compare the 90%, 95%, and 99% confidence intervals for the true mean visit duration.

Before opening the solution, compute the shared standard error, then give all three intervals and explain why their centres agree while their widths differ.

Show Solution

Confidence		Margin of Error	Interval
90%	1.645	minutes	(50.17, 54.63) minutes
95%	1.96	minutes	(49.74, 55.06) minutes
99%	2.576	minutes	(48.91, 55.89) minutes

Key insight: All three intervals share the same center . Higher confidence demands a wider interval: you pay for certainty with precision. Using unrounded margins, the 99% interval is about 57% wider than the 90% interval (widths about 6.99 and 4.46 minutes).

Example 4 — Determining Required Sample Size (Application Twist)

An airline wants to estimate the true mean baggage weight per passenger to within kg at 95% confidence. From historical data, kg. How many passengers must be sampled?

Before opening the solution, choose the planning formula, compute the raw sample-size requirement, state the minimum whole-number sample, and explain why the rounding direction matters.

Show Solution

Interpretation: The airline needs to sample at least 62 passengers so that a 95% interval for the mean has a margin of error of at most ±15 kg.

Check your intuition: What if the airline wanted the same margin of error with 99% confidence?

Going from 95% to 99% confidence requires jumping from 62 to 107 passengers, 73% more data, for the same precision.

Section 5: Guided Practice

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Work through each problem step by step. Every step gives feedback as soon as you commit, and wrong answers explain what went wrong.

Problem 1 — Constructing a Confidence Interval (C2 + C3)

Problem 2 — May the Report Print This? (C4)

Sort into groups

May the report print this sentence?

A café's 95% confidence interval for the mean wait from order to pickup is (3.90, 4.50) minutes. Sort each sentence: is it a correct reading, or which mistake does it make?

Drag an item into its group, or select it and then choose the group. On a keyboard: select an item and press the group’s number (1–4); 0 sends it back to Unsorted. Each item also has its own “move” buttons.

Unsorted

Problem 3 — Determining Required Sample Size (C6)

Problem 4 — Precision vs. Confidence Trade-off (C5)

Complete the table

One sample, three confidence levels

Fill in every empty cell, then check the whole table once. The cells depend on one another.

A colleague looks at a table like this one and argues: “We should always use 99% confidence, because more is always better.” Which response is most defensible?

Section 6: Independent Practice

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Less support now: commit to each answer before you look at any feedback.

Problem 1 — Can This Lesson’s Interval Be Used? (C2)

Problem 2 — Constructing a Confidence Interval (C2 + C3)

Problem 3 — One Commuter, or the Average Commuter? (C4)

Structural Difference

A range for individuals, or an interval for the mean?

Identify what key word or condition separates Case A from Case B, then select the method for each.

Scenario A

Commute times in a city have μ = 34 min and σ = 12 min, with no extreme skew. Between which two times do the middle 95% of individual commutes fall?

Select method:

Scenario B

A random sample of 64 commuters in the same city averages x̄ = 34 min; σ = 12 min is known. Give a 95% confidence interval for the mean commute time.

Select method:

Problem 4 — Does the Mean Clear the Threshold? (C4)

Build the Conclusion

Read the interval against the threshold

Choose a phrase for each blank so that the sentence states the correct conclusion.

Problem 5 — Sample Size and the Quadruple Rule (C5 + C6)

Problem 6 — Diagnose Flawed Interval Work (C2–C6)

Review Feed

Problem 7 — Full Synthesis: Sodium Intake (C2 + C3 + C4 + C6)

Mixed Review — Normal Probability (L-INF-1)

Mixed Review — Sampling Distribution (L-INF-1)

Section 7: Mastery Check

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

No hints, no scaffolding: one cold attempt per idea. Steps open only if you ask for them after your own attempt.

Question 1 — Feynman Test

A quality report says: “The estimated mean fill volume is 502 mL, with a margin of error of ±1.5 mL at 95% confidence.” Explain to a friend what the margin of error measures, how it is built from and the standard error, and what the 95% says about the interval procedure.

Write a concise explanation in plain language on paper or in your notes. Then compare your wording with the model answer.

Compare with a model answer

The reported interval is mL. Its margin of error is the critical value times the standard error: . The standard error describes how much sample means vary from sample to sample; reaches far enough on either side for the 95% procedure.

The 95% means that if the company repeatedly drew random samples and built intervals with this method, about 95% of those intervals would contain the fixed true mean fill volume. It does not mean that individual bottles have a 95% chance of falling in this interval or that the fixed mean has a 95% probability of being inside this particular interval.

Question 2 — Cold CI Construction

Question 3 — What May the Report Say?

Question 4 — Decide With the Interval

Question 5 — Plan the Sample

Self-Assessment

How confident are you with the concepts from this lesson?50% confident

Still confusedReady for the Boss Fight

Section 8: Boss Fight

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Choose one of two equally demanding missions. Neither tells you which method to use: choosing the tool is part of the fight. Each starts with your own complete attempt; open the steps only if you need to find where it went wrong. Lose, and you get a fresh rematch with new numbers.

🎲 The Replication Bet

Back to the opening bet: how often does a colleague’s repeat study land inside your interval?

🏗️ The Architect

A planning office wants a precise estimate on a fixed budget. Can it afford the study it wants?

🎲 The Replication Bet

Then, check it by simulation. Before you press the button, predict how many of 100 replication means will land inside the first study’s interval.

Predict: how many of 100 replications will land inside? Then press Replicate.

Figure 7: The replication bet. Your study timed n = 50 coffee orders (σ = 1.1 min known) and built a 95% interval for the mean wait. Each orange dot is the sample mean from one exact repeat of your study by a colleague. Solid dots landed inside your interval; faded rings landed outside. Try several first studies and watch the running average.

Then, on paper: go back to the bet you wrote in the Introduction. In two or three sentences, explain why the answer is below the confidence level, and what the confidence level does promise.

Compare with a model answer

For two studies of the same size, a repeat study’s mean lands inside a 95% interval only about 83% of the time on average. My interval is centred on my own , which already missed by some amount; the colleague’s misses independently. The difference of the two means has spread , so the 1.96·SE reach is only about 1.39 of those spreads. The 95% promises something else: about 95% of intervals built this way capture the fixed true mean .

🏗️ The Architect

Then, on paper: write two or three sentences the planning office could read. Say whether the study is affordable as specified, and if not, which requirement you would revisit first and why.

Compare with a model answer

The required sample comes from , rounded up, and the cost is that sample times the cost per resident. If the cost is within the cap, the office can run the study as specified. If not, an honest plan changes an input and recomputes: a slightly larger target margin cuts the sample by the square of the change, while a lower confidence level raises the long-run miss rate. It never rounds down or simply claims more precision than the data can buy.

Section 9: Challenge Problems

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Ready for more? These optional problems go beyond the lesson objectives. They do not announce the formula or sequence to use. Decide what the situation demands, commit one final answer, and then compare your reasoning with the worked solution.

Challenge 1 — Precision Has a Price

Challenge 2 — Build a Critical Value from Scratch

Challenge 3 — Plan with Imperfect Information

Section 10: Solutions Reference

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Complete, step-by-step solutions for all problems in Sections 5–9 are available on the solutions page. Solutions include worked arithmetic, common mistakes to watch for, and interpretation guidance.

View Full Solutions →

If you’re stuck: Re-read the relevant Core Concept in Section 3, then find the Worked Example that maps to it (Example 1 builds an interval, Example 2 adds a decision, Example 3 compares widths, Example 4 plans a sample size). The solutions page shows the reasoning behind every step, not just the final answer.

Quick-Reference Formulas

Confidence Interval for (Large Sample):

If is unknown, use the t procedure developed in L-INF-3.

Margin of Error ():

Required Sample Size: (The ceiling means always round the raw requirement up to the next whole number.)

Confidence Level	Rounded Critical Value ()
90%	1.645
95%	1.96
99%	2.576