You’ve just finished INF-1, where you learned that a sample mean is itself a random variable — if you took a different sample, you’d get a different value. The Central Limit Theorem tells us is approximately normally distributed around the true population mean .
But here’s the real-world problem: you almost never know. You have one sample, one , and you need to say something useful about the unknown truth. How close is your estimate? How uncertain are you?
A confidence interval is the answer. Instead of reporting a single number (“the mean delivery time is 31.4 minutes”), you report a range: “I am 95% confident the true mean delivery time is between 29.8 and 33.0 minutes.” That range quantifies your uncertainty in a statistically principled way.
This lesson teaches you to construct, interpret, and use confidence intervals for a population mean when the sample is large. Everything here builds on the CLT from INF-1 — if that lesson is solid, this one will feel natural.
After this lesson, you will be able to:
Construct a confidence interval for using the formula
State the correct frequentist interpretation of a confidence interval and identify the most common misinterpretation
Choose the appropriate critical value for 90%, 95%, and 99% confidence levels
Explain how confidence level, sample size, and population variability affect interval width
Determine the minimum sample size needed to achieve a desired margin of error using
This is Lesson INF-2. The next lesson (INF-3) extends this to the case where is unknown and the t-distribution is required. Master the framework here first.
Section 2: Prerequisites
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Confidence intervals are a direct application of the Sampling Distributions you studied in INF-1.
From INF-1: Standard Error (SE).. This measures the typical “error” or distance between the sample mean and the population mean .
From INF-1: The CLT Guarantee. If , the sampling distribution of is approximately normal, regardless of the population’s shape.
Critical Z-Values: For a given confidence level, is the number of standard errors you must go out from the mean to capture that percentage of the distribution.
Normal Table Fluency: You must be able to find a z-score given an area (inverse lookup).
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?
Success Factor:
If you cannot calculate the Standard Error () correctly, your confidence interval will be too wide or too narrow. Ensure you are using and not just in the denominator.
Retrieval Warm-up — from earlier lessons
The heights of adult women in a city are normally distributed with cm and cm. Using a z-table where and , what fraction of women are between 160 and 172 cm tall?
You take a random sample of women from the same population ( cm, cm). What is the standard error of the sample mean, and what does it tell you?
Section 3: Core Concepts
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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.
C2 — Constructing the Confidence Interval
The confidence interval for is built by going standard errors out from in both directions:
Confidence Interval for μ (Large Sample, σ Known)
which gives the interval:
Conditions required:
Random sample (to justify probability calculations)
or the population is approximately normal (for the CLT to apply)
is known (if unknown and , substitute — see INF-3 for the exact t-distribution approach)
Figure 3: The perspective flip. Panel A (INF-1 view) shows the sampling distribution centred at the known μ — x̄ is the random quantity. Panel B (INF-2 view) shows the same ±z*·SE geometry centred at the observed x̄ — μ is the fixed target. Click New sample to draw a random x̄ and observe that both panels always agree: the same algebraic condition determines whether the middle-region is satisfied.
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.
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
Critical value
90%
0.05
1.645
95%
0.025
1.96
99%
0.005
2.576
These three values are worth memorizing — they appear in every inference problem through this module.
Figure 4: The anatomy of z*. The blue region contains the middle C% of the standard normal distribution; the orange tails together hold the remaining (1−C)%. 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.
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 — The Correct (Frequentist) Interpretation
This is the most commonly misunderstood concept in elementary statistics. Read it carefully.
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 — it doesn’t have a probability of being anywhere. 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.
Accepted shorthand. In textbooks, reports, and the rest of this course you will also read: “We are 95% confident that the true mean lies between [L] and [U].” This phrasing is correct and widely used — it is shorthand for the longer procedure-language statement above. Read it that way; do not interpret it as assigning a probability to the fixed constant .
The visualization below makes this concrete: each horizontal bar is one confidence interval from a different random sample. Green bars capture ; red bars miss it. Watch what fraction are green as you draw more samples — it converges to your stated confidence level.
See it for yourself. Draw confidence intervals repeatedly. Each interval comes from a different random sample from the same population (μ = 50). Notice that most intervals capture the true mean — but some miss. The confidence level tells you what fraction of all possible intervals would succeed.
Figure 2: Each horizontal bar is one confidence interval built from a different random sample (μ = 50, σ = 10, n = 36). Green bars capture the true mean; red bars miss it. The running tally shows how often the procedure succeeds — watch it converge to the stated confidence level.
Figure 1: CI Coverage Explorer — each horizontal bar is one confidence interval built from a different random sample. Green bars capture the true population mean μ (orange dashed line); red bars miss it. Watch the running tally approach the stated confidence level as you draw more samples.
Use the controls to explore: What happens to the width of bars when you increase the confidence level from 90% to 99%? What happens to the capture rate when you switch from 95% to 90%? Notice that individual intervals have no probability attached — the 95% is a property of the procedure, visible only across many repetitions.
Figure 6: Both panels share the same x-axis. Panel A shows where individual observations fall — the shaded band is ±z*·σ ≈ ±20 units wide and doesn't change. Panel B shows where sample means fall — the shaded band is ±z*·SE = ±z*·σ/√n, which shrinks as you increase n. A 95% CI captures 95% of sample means, not 95% of individual data points.
The most common error in this lesson: “The probability that falls in the interval is 95%.” This is wrong. The probability language applies to the interval (a random quantity, because it depends on your sample), not to (a fixed constant). Say instead: “We used a procedure that captures 95% of the time.”
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.
Figure 5: All three intervals share the same centre x̄. Drag the n slider to watch every bar shrink together — doubling n reduces each width by 29%; quadrupling n halves all widths. Drag σ to see how population variability scales all three proportionally. The relative widths (gold < blue < purple) never change because they depend only on z*.
The only factor you control in practice is . You can’t change (it’s a property of the population), and lowering the confidence level to get a narrower interval makes your inference weaker.
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 7: The cost of precision. Drag the orange dot (or click anywhere on the chart) to set your desired margin of error E. The gold dot shows what happens if you halve E — the required sample size approximately quadruples, 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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Example 1 — Constructing a 95% CI (Fully Worked)
A coffee shop times the wait from order to pickup. They know from years of records that minutes. A random sample of orders yields minutes. Construct a 95% confidence interval for the true mean wait time .
Step 1 — Verify conditions:
Random sample: ✓ (stated)
: ✓ CLT applies
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 3.89 and 4.51 minutes. This means we used a procedure that captures the true mean in about 95% of samples drawn this way.
Example 2 — Using the Sample SD When σ is Unknown (Partially Scaffolded)
A health researcher measures the resting systolic blood pressure of a random sample of adults. The sample yields mmHg and mmHg. The population SD is unknown. Construct a 90% confidence interval.
Note on using s: Since is unknown, we substitute in the SE formula. This is a large-sample approximation valid because . The exact method (t-distribution) will be covered in INF-3.
Before seeing the solution: which value applies to 90% confidence — 1.645, 1.96, or 2.576? What changes in the formula when we use instead of ?
Step 1 — Conditions:, random sample, unknown but available — large-sample approximation valid.
Step 2 — Critical value:
Step 3 — SE and margin of error (using s):
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.
Example 3 — Comparing Widths Across Confidence Levels (Minimally Scaffolded)
A sample of measurements yields and the population SD is . Compute and compare the 90%, 95%, and 99% confidence intervals.
Show Solution
Confidence
Margin of Error
Interval
90%
1.645
(50.17, 54.63)
95%
1.96
(49.74, 55.06)
99%
2.576
(48.90, 55.90)
Key insight: All three intervals share the same center . Higher confidence demands a wider interval — you pay for certainty with precision. The 99% interval is about 57% wider than the 90% interval (width 7.00 vs. 4.46).
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?
Show Solution
Interpretation: The airline needs to sample at least 62 passengers to be 95% confident that their sample mean is within ±15 kg of the true population mean.
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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Work through each problem step by step. The dropdowns give immediate feedback — wrong answers explain what went wrong.
Problem 1 — Constructing a Confidence Interval (C2 + C3)
Coffee shop. Customer satisfaction ratings (0–10 scale) have . A random sample of customers gives . Construct a 95% CI for the true mean rating.
Step 1: What is the standard error?
Step 2: What is the margin of error for a 95% CI?
Step 3: Which interval is correct?
Exam scores. A standardized exam has known points. A random sample of students gives points. Construct a 99% CI for the true mean score.
Step 1: Standard error?
Step 2: Margin of error for 99% CI?
Step 3: The 99% CI is:
Delivery times. A logistics company has minutes for delivery times. A random sample of deliveries gives minutes. Construct a 90% CI for the true mean delivery time.
Step 1: Standard error?
Step 2: Margin of error for 90%?
Step 3: The 90% CI is:
Resting heart rate. For adult males, bpm. A random sample of gives bpm. Construct a 95% CI.
Step 1: Standard error?
Step 2: Margin of error for 95%?
Step 3: The 95% CI is:
Battery life. A manufacturer knows hours for a battery model. A random sample of batteries gives hours. Construct a 99% CI for the true mean battery life.
Step 1: Standard error?
Step 2: Margin of error for 99%?
Step 3: The 99% CI is:
Problem 2 — Interpreting Confidence Intervals (C4)
A researcher computes a 95% CI for mean daily step count: (8,240, 9,760) steps. Which statement correctly interprets this interval?
A 90% CI for mean exam score is (71.3, 76.7). A student says: “I’m 90% sure the true mean is in this interval.” What’s wrong with this statement?
After computing a 95% CI of (14.2, 17.8) for mean waiting time (minutes), a manager says: “95% of customers wait between 14.2 and 17.8 minutes.” Is this correct?
A 99% CI for mean temperature is (36.1°C, 36.9°C). Which statement is most accurate?
True or false: “Once we compute a 95% CI, the probability that μ is inside it is either 0 or 1 — we just don’t know which.” Explain.
Problem 3 — Determining Required Sample Size (C6)
Problem 4 — Precision vs. Confidence Trade-off (C5)
A sample of measurements yields with known .
Two confidence intervals are constructed from this sample: one at 90% confidence, one at 99%.
Step 1: Which interval is wider?
Step 2: Which interval gives a more precise estimate of ?
Step 3: A colleague argues: “We should always use 99% confidence — more is always better.” Which response is most defensible?
Show computed intervals
Width of 90% CI: . Width of 99% CI: . The 99% CI is 2.54 units wider — about 57% wider for the same data.
Section 6: Independent Practice
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No dropdowns — work through each problem fully, then check the solution.
Problem 1 — Large-Sample CI Using s When σ is Unknown (C2 + C3)
Study hours. A researcher surveys university students. The sample yields hours/week studying and hours. The population SD is unknown. Construct a 95% CI for the true mean study time.
Show Solution
Conditions:. Use large-sample approximation with in place of .
Interpretation: We are 95% confident the true mean weekly study time is between 16.8 and 19.6 hours.
Commute time. A city surveys residents. Sample: minutes, minutes. Construct a 90% CI for the mean commute.
Show Solution
Electricity bill. A utility samples households. Sample: = $142.50, = $28.00. Construct a 99% CI for the mean monthly bill.
Show Solution
Hospital stay. A hospital records patient stays. Sample: days, days. Construct a 95% CI for the mean length of stay.
Show Solution
Student height. A sample of students yields cm and cm. Construct a 90% CI for the mean height.
Show Solution
Problem 2 — Constructing a Confidence Interval (C2 + C3)
Problem 3 — Sample Size and the Quadruple Rule (C5 + C6)
Problem 4 — Find the False Statement (C4 + C5)
A researcher reports a 95% CI of (42.1, 47.9). Four colleagues make the following statements. Which one is false?
Show the Statements and Solution
A: “The margin of error is 2.9.”
B: “The sample mean used was 45.0.”
C: “There is a 95% probability that μ = 45.0.”
D: “If we took 1,000 more samples and built 1,000 more CIs this way, about 950 of them would contain μ.”
False statement: C. μ is a fixed constant, not a random variable. It cannot have a “95% probability” of equaling any value. Statements A, B, and D are all correct: A — half-width = (47.9 − 42.1)/2 = 2.9 ✓; B — center = (42.1 + 47.9)/2 = 45.0 ✓; D — correct description of long-run coverage ✓.
A 99% CI for mean daily calories is (1,840, 2,160). Which statement is false?
Show the Statements and Solution
A: “The width of this interval is 320 calories.”
B: “99% of people in the study consume between 1,840 and 2,160 calories.”
C: “The procedure used to build this CI captures μ in about 99% of all samples.”
D: “A 95% CI computed from the same data would be narrower than this interval.”
False statement: B. A CI for μ does not describe where individual observations fall — it describes where the population mean is. Eating habits of individuals span a far wider range than this CI. Statement D is true: lower confidence → smaller z* → narrower interval.
A 95% CI computed from is (18.0, 22.0). A second CI is computed from with the same and . Which statement is false?
Show the Statements and Solution
A: “The second CI will be narrower.”
B: “Quadrupling n halves the margin of error.”
C: “The second CI has a 95% capture rate, just like the first.”
D: “The second CI is more precise, so it has a higher probability of capturing μ than the first.”
False statement: D. Both intervals are 95% CIs — they both have the same coverage probability of 95%. Narrower means more precise, not more likely to capture μ. The confidence level is set by z*, not by sample size.
Which statement about the margin of error is false?
Show the Statements and Solution
A: “Doubling σ doubles E (all else equal).”
B: “Quadrupling n halves E.”
C: “Raising the confidence level from 95% to 99% increases E.”
D: “Doubling n halves E.”
False statement: D. Doubling n replaces with , so E shrinks by factor — a 29% reduction, not a halving. To halve E, you must multiply n by 4.
A 95% CI is reported as (55.2, 64.8). A skeptic says: “Since this interval doesn’t contain 50, we can be 95% confident that μ ≠ 50.” Another says: “The interval means μ is likely near 60.” Which comment is more problematic, and why?
Show Solution
The first comment is more problematic. The statement “95% confident that μ ≠ 50” confuses hypothesis testing with CI construction, and overextends what confidence intervals prove. A CI not containing 50 does provide some evidence against μ = 50 (it’s equivalent to rejecting H₀: μ = 50 at 5% significance), but saying “95% confident μ ≠ 50” misstates both the CI’s meaning and the conclusion.
The second comment — “μ is likely near 60” — is a reasonable informal gloss on the CI. It’s not precisely worded but captures the correct direction: values near the center of the CI are most consistent with the data.
Problem 5 — Full Synthesis: Sodium Intake (C2 + C3 + C4 + C6)
A nutritionist is studying daily sodium intake in a university population. From a national study, mg (assumed applicable). A random sample of students yields mg/day.
Part (a): Construct a 95% CI for the true mean daily sodium intake.
Part (b): The US recommended daily limit is 2,300 mg. Based on your CI, is there statistical evidence that the mean exceeds the limit?
Part (c): The nutritionist wants to cut the margin of error from Part (a) in half. Without computing, first predict roughly what sample size would be required. Then calculate the exact and compare to your prediction.
Show Solution
Part (a):
Part (b): The entire CI lies above 2,300 mg — the lower bound is 2,321 mg, which exceeds the limit. This provides statistical evidence that the mean sodium intake exceeds the recommended level. The data are inconsistent with at the 95% confidence level.
Part (c):
Prediction: Since , halving requires multiplying by . Predicted: students.
Exact calculation: The target margin of error is mg.
The exact answer (201) matches the prediction almost perfectly — the tiny discrepancy comes from rounding to 88.7. Halving the margin of error quadruples the required sample size. This is the most important practical consequence of the relationship.
Mixed Review — Retrieval from Earlier Lessons
These problems draw on concepts from earlier in the course. Attempting them without re-reading prior lessons is the point — retrieval practice strengthens long-term memory more than re-reading.
Review Problem 1 — Normal Probability (PR-6)
A machine fills cans with juice. Fill volumes are normally distributed with mL and mL. A consumer protection standard requires that fewer than 5% of cans contain less than the labelled 350 mL. Using the fact that , does this production process meet the standard?
Show Solution
Standardize the threshold:
The process does not meet the standard: 10.56% of cans are underfilled, well above the 5% threshold. The machine would need to increase its mean fill to at least mL to bring the underfill rate below 5%.
Review Problem 2 — Standard Error and Sampling Distribution (INF-1)
Exam scores at a large university are known to have points and points. A professor randomly selects a class of students.
(a) What is the standard error of the class mean ? (b) What does the CLT guarantee about the shape of the sampling distribution of ? (c) What is ? (Use .)
Show Solution
(a) points.
(b) Since , the CLT guarantees that is approximately normally distributed around , regardless of the shape of the score distribution in the population.
(c). So . About 3.5% of random classes of this size would average above 75 points.
Section 7: Mastery Check
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No hints, no scaffolding — these questions measure genuine understanding.
Question 1 — Feynman Test
A friend who hasn’t taken statistics asks: “What does it mean when a news article says ‘margin of error ±3 points at 95% confidence’? How is that calculated and what does the 95% actually mean?”
Write your explanation in plain language (complete sentences, no jargon).
0 / 500
See a model answer
The margin of error (±3 points) tells you how far the reported number might be from the true population value. It’s calculated as: critical value × (population SD / √sample size). The critical value for 95% confidence is 1.96 — a number from the normal distribution.
The “95% confidence” means: if this poll were repeated many times using the same method, about 95 out of every 100 resulting intervals would contain the true value. It does not mean “there’s a 95% chance the true value is in this specific interval” — the true value is fixed; it’s either in there or it isn’t.
Question 2 — Light Bulb CI
A random sample of light bulbs from a production line has hours with known hours.
Step 1: The standard error is:
Step 2: The 95% margin of error is:
Step 3: The correct 95% CI is:
Show full solution
We are 95% confident the true mean bulb life is between 1,195 and 1,235 hours.
Question 3 — Error Analysis
A student is asked to construct a 95% CI for a mean with , , . They write:
“The 95% CI is .”
“I am 95% confident that μ is in this interval.”
There is one definite error in this work. Identify it. Then consider whether the final sentence also represents a problem.
Show Analysis
Error (computational): The student plugged in directly instead of the standard error . The correct margin of error is , giving the interval (82.04, 85.96) — much narrower. Using in place of SE inflates the interval by a factor of , making it nearly 10 times too wide.
On the final sentence: “I am 95% confident that μ is in this interval” is the standard accepted shorthand used in every major statistics textbook — it is not an error. The stricter frequentist statement is: “We used a procedure that captures μ in 95% of all samples constructed this way.” Both phrasings are acceptable; the shorthand should not be read as claiming μ is random or that the probability is 95% once the interval is in hand. The only real error here is the computation.
Self-Assessment
How confident are you with the concepts from this lesson?
Still confusedReady for the Boss Fight
Section 8: Boss Fight
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You’ve practiced individual concepts. Now use all of them at once — under pressure. Choose your path:
🔬 The Analyst
You have a dataset. Build and interpret confidence intervals at two levels, then determine the sample size needed for a tighter estimate.
🏗️ The Architect
You’re designing a study before data collection. Compare the cost of 95% vs. 99% confidence and present a justified recommendation.
🔬 Path A: The Analyst
A city’s public health department collects data on daily physical activity (steps) from a random sample of adults. From national fitness tracking data, steps (assumed applicable to this population). The sample yields steps/day.
Task A1: Construct both a 95% and a 99% CI for the true mean daily step count.
Task A2: The World Health Organization recommends 10,000 steps/day. Does your 95% CI provide evidence that the mean in this city exceeds 10,000 steps? Explain carefully.
Task A3: The department wants to reduce the margin of error to at most 1,500 steps while maintaining 95% confidence. What sample size is required?
Task A4: How does your answer to A3 change if they use 99% confidence? Comment on the cost trade-off.
Show Full Solution
SE: steps
1. Confidence intervals:
2. Interpretation: Both CIs have lower bounds well above 10,000 steps (the WHO recommendation). The data provide very strong evidence that the mean in this city exceeds 10,000 steps — this result holds at both the 95% and 99% confidence level.
3. Sample size for E ≤ 1,500 at 95%:
4. At 99% confidence:
Going from 95% to 99% requires 153 vs. 89 participants — 72% more data collection effort — for the same precision. Whether this is worthwhile depends on the stakes of an incorrect conclusion about public health policy.
Reflection: What was the most challenging part of this analysis? Was it the initial setup or the final interpretation?
🏗️ Path B: The Architect
You are a health economist designing a study to estimate the mean annual healthcare cost per patient for a rare chronic disease. Costs in this disease category are highly variable: from published literature, = $2,500. Your stakeholder (a government health agency) wants the estimate to be within = $300 of the true mean.
Task B1: Calculate the required sample size for 95% confidence. Calculate it again for 99% confidence.
Task B2: Patient recruitment costs $120 per participant. Compute the total recruitment cost for each confidence level.
Task B3: The agency pushes back: “We’re fine with 90% confidence if it saves money.” Compute the required and total cost at 90%.
Task B4: Write a two-paragraph recommendation: which confidence level do you advise, and why? Consider both statistical and practical factors.
Show Full Solution
1. Required sample sizes:
2. Recruitment costs:
95%: 267 × 32,040**
99%: 461 × 55,320**
3. At 90% confidence:
90%: 188 × 22,560**
4. Recommendation (model answer):
The 95% confidence level strikes the right balance between statistical rigor and resource efficiency. It requires 267 patients and 9,480 but leaves a 10% chance that the interval misses the true mean; for government policy decisions with real budget implications, that miss rate may be unacceptably high.
The 99% CI, while most robust, costs nearly $23,000 more than the 95% option for a marginal improvement in confidence. Unless the agency has specific regulatory requirements for 99% precision, the 95% CI is the defensible standard recommendation. If patient recruitment is less expensive than estimated, reassess.
Reflection: What was the most challenging part of this analysis? How would you apply this design approach to another problem?
Section 9: Challenge Problems
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Optional stretch. These go beyond the lesson objectives. Attempt them when you feel confident with the core material.
A study requires a 95% CI with margin of error units. The population SD is . A grant limits the study to 100 participants. Is the target achievable? If not, what margin of error can they realistically achieve with 100 participants?
Show Solution
Required n:. With only 100 participants, the target is not achievable.
Achievable E with n = 100: units. The best they can achieve is a margin of error of about 5.9 — modestly above the 5-unit target.
Target: 90% CI with , , maximum . Is the target achievable?
Show Solution
Required n:. Yes — 68 ≤ 70, so the target is achievable with 70 participants.
Challenge 3 — Why Halving the Margin of Error Requires Quadrupling n
The formula shows that margin of error and sample size are connected through a square root. This challenge derives the exact relationship algebraically.
(a) Solve the margin-of-error formula for .
Show
(b) Suppose a researcher wants a new margin of error (half the original), keeping the same confidence level and population. Write the expression for in terms of .
Show
(c) Square both sides of your answer to part (b). Express in terms of , and state the general rule.
Show
General rule: Since , reducing by a factor of requires multiplying by . Halving () quadruples . Reducing by one-third () requires nine times the sample. This is why precision improvements quickly become expensive.
Challenge 4 — Where Does z* = 1.96 Come From?
Most students treat as a memorised fact. This challenge shows exactly how it is derived from the standard normal distribution.
(a) A 95% confidence interval captures the middle 95% of the standard normal — the region from to . What is the total area in the two tails? What is the area in each individual tail?
Show
Total tail area . By symmetry of the normal distribution, each tail contains .
(b) Since the right tail has area 0.025, what is the cumulative area to the left of ?
Show
All area to the left of = left tail + middle = .
(c) A standard normal table gives . What does this confirm, and how would you find for 99% confidence using the same method?
Show
Since , the value is exactly the z-score where the left-tail cumulative area equals 0.975 — confirming that the middle 95% of the standard normal lies between and .
For 99% confidence: each tail has area , so . Looking up 0.995 in the z-table gives .
General formula: For a confidence level, . The three standard values are just inverse-normal lookups at 0.950, 0.975, and 0.995.
Challenge 5 — Confidence Intervals and Hypothesis Tests
There is a deep connection between confidence intervals and hypothesis tests: a 95% CI contains exactly those values of that would not be rejected by a two-sided hypothesis test at significance level .
A 95% CI for a mean is computed as (38.2, 45.8).
a. Without computing a p-value, what would you conclude about the hypothesis ?
b. What about ?
c. Explain why this equivalence holds — why does the CI tell you about hypothesis test outcomes?
Show Solution
(a) falls inside (38.2, 45.8), so a two-sided test of at would fail to reject — the data are consistent with this value.
(b) falls outside the CI, so the test would reject — the observed is too far from 37 to be explained by chance at the 5% level.
(c) The CI is built using the condition , i.e., . The hypothesis test rejects when . These are exactly complementary — the CI contains the values that wouldn’t trigger rejection. The 95% CI and the 5% two-sided test are two views of the same mathematical result.
Challenge 6 — Conservative Sample Size When σ is Unknown
The sample size formula requires knowing . When is unknown before data collection, a common conservative rule uses — a rough estimate based on the fact that roughly 95% of a normal distribution falls within 2 SDs of the mean, so the full range ≈ 4σ.
A researcher wants to estimate mean study hours per week for university students. Students seem to range from about 0 to 40 hours per week. She wants a 95% CI with hours.
a. Estimate using the range/4 rule.
b. Compute the required sample size.
c. Why is this called “conservative”? What is the risk of using too small an estimate of σ?
(c) “Conservative” means the formula tends to overestimate n — it’s safer to collect too many observations than too few. If is overestimated, you’ll collect more data than needed but the target E will still be met. If is underestimated, your final CI will be wider than planned — the study fails its precision objective. Overestimating σ wastes some resources; underestimating it means the study fails entirely.
Section 10: Solutions Reference
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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.
If you’re stuck: Re-read the relevant Core Concept in Section 3, then find the Worked Example that maps to that concept (e.g., Example 1 maps to Concept 1). 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 and , substitute for .
Margin of Error (MOE):
Required Sample Size:(Always round up to the next whole number)