L-INF-3: Solutions — Confidence Intervals for a Population Mean (σ Unknown)

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.

Critical values shown in the lesson tables are rounded to three decimal places. Accordingly, fixed decimal critical values, margins, and interval endpoints below are approximate unless the arithmetic is exact from the stated values.

Problem 1 — Build a t-Interval, Step by Step

This problem regenerates. For the displayed values:

1. Compute the degrees of freedom.

2. Find in the excerpt: the row (not the row , which the excerpt also shows) and the requested confidence column.

3. Compute the standard error.

4. Compute the margin of error.

5. Center the interval at the sample mean.

In this problem is , , or , so is an integer. Keep the full value of through the calculation and round only the requested answer.

Problem 2 — What Does the Table Header Mean?

Select both tails. A 95% interval leaves outside , split as in each tail. The header “0.025 (1)” names one tail; “0.05 (2)” names the two tails together. With the cut-offs are .

Problem 3 — z, t, or No Interval Yet?

Ask two questions in order:

1. Do the sampling and the shape support a mean interval? A random, independent sample is required. For small samples the dot plot should show one mound with no extreme outliers; a long tail with a far value, a cluster of people working together, or self-selected volunteers each rule an interval out for now.
2. Is known from outside the sample? Records, a specification, or a long calibration history supply : use z. If the only spread is the sample’s , use t, at any sample size. Old or “historical” data do not make known.

Problem 4 — What Changes the Method?

Case A gives the population standard deviation, so it uses z. Case B gives only the sample standard deviation, so it uses t. Everything else is identical: the source of the spread is the single fact that changes the method.

Problem 5 — Same Spread, Three Sample Sizes

For each row, read at in the 95% column, then compute

For example, with : gives and ; gives and ; gives and . Both ingredients shrink as grows, and the t margin approaches the z-based margin from above.

Which margin to report: the t margins. The spread was estimated by , so the z-based margins claim more precision than the data support. The gap is largest for the smallest sample.

Problem 6 — On Your Own, with Steps on Request

Same method as Problem 1: , then from the excerpt, then , , and .

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.

Problem 1 — Margin of Error from Sample Data

The values regenerate, and is not a perfect square. Read at , then

using the unrounded standard error. The main wrong methods are:

• using when is unknown;
• reading the t-table at instead of ;
• multiplying by instead of by .

Problem 2 — Choose the Distribution, Then Compute

The generator has two possible branches.

Branch A — only the sample’s is available

Use t with , because is estimated by .

Branch B — calibration records give

Use z, because is known from outside the sample.

In both branches the independent observations and approximately normal population support a mean interval; what is known about the spread selects the critical value.

Problem 3 — Read the Interval Against a Threshold

1. The interval uses because the population standard deviation was estimated from the sample. Sample size and a normal-looking population do not choose between z and t.
2. Compare the value with the endpoints. If it lies below the whole interval, the data show the mean is above it; if above the whole interval, the data show the mean is below it; if inside, plausible means lie on both sides and the data cannot tell.

Problem 4 — Diagnose a Confidence-Interval Report

The generated report contains one of five possible first errors. Each error is carried forward consistently, so the first wrong step is the only one to name.

Error	Correct diagnosis
Uses z because of sample size	Unknown calls for t; sample size does not make known.
Uses	Use because estimating the sample mean constrains one deviation.
Uses	Use .
Assigns probability to after computing the interval	is fixed; confidence describes the repeated procedure.
Skips sampling and shape checks	t does not repair a non-random, dependent, or badly shaped small sample.

Problem 5 — Estuary Salinity Study

Part (a) — Unknown- salinity study

The sample has , ppt, and ppt. With , the 95% column gives .

Using the unrounded standard error:

Part (b) — Matched known- comparison

A long-run model of the same estuary supplies known ppt for the same readings.

The t interval from Part (a) is wider because . In Part (a), that extra width is justified because is estimated; in the matched analysis, known justifies z.

Part (c) — Why the matched studies use different intervals

In Part (a), is unknown and replaced by the sample standard deviation . The t critical value accounts for the extra sampling uncertainty in that estimate. In Part (b), an external model supplies known , so z is justified instead.

Part (d) — Four times as many observations

Holding , , and the confidence level fixed, changing from to affects both ingredients in the margin:

• is halved because multiplying by four doubles .
• increases from 10 to 43, so the matching decreases toward the corresponding z critical value.

Both changes narrow the interval under the stated controlled comparison.

Mixed Review — Retrieval from L-INF-1 and L-INF-2

For the L-INF-1 sampling-distribution review:

Then read the requested right-tail area from the reference values displayed in the step.

For the L-INF-2 known- review:

A valid interpretation says that we are 95% confident the interval captures the true population mean. It does not describe individual observations or assign a 95% probability to a fixed .

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.

Question 1 — Feynman Test

When is unknown, we estimate it with , and that estimate varies from sample to sample. Samples whose comes out too small would give intervals that are too short. The t-distribution accounts for this extra uncertainty with heavier tails and a larger critical value at finite . Multiplying the standard error by the larger produces a wider interval. The extra width is honest uncertainty, not a defect.

Question 2 — Cold Interval

The question does not name the procedure.

1. Confirm that the independent observations and stated shape evidence support a mean interval.
2. If the only spread is the sample’s , choose t and read at . If the instrument’s calibration record gives , choose z.
3. Compute , then = critical value × , then .
4. Interpret the interval as plausible values for the population mean, not as a range for individual values.

Question 3 — Which Procedure?

Check the license first (random, independent, shape suited to ), then what is known about the spread: known → z; estimated by → t; a failed condition → no mean interval yet.

Question 4 — Why the Shortcut Falls Short

The sample standard deviation varies. When it comes out smaller than , the interval is too short and misses . The standardized mean follows a t-distribution with , so a larger is needed to restore the advertised coverage. The population being normal rules out the “fewer than 30” explanation.

Question 5 — Decide With the Interval

Compare the value with the endpoints: below the whole interval → the mean is above it; above the whole interval → the mean is below it; inside → the data cannot tell.

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.

Path A — Two Analysts, One Sample

1. Method. If only the sample’s is available, Analyst Z’s interval treats as if it were : the right interval uses with . If the supplier’s records give , Analyst Z is right.
2. Critical value. Read at in the 95% column (or use when is known).
3. Interval. .
4. Verdict. Compare the contract value with that interval. When the value falls between the z and t endpoints, Analyst Z’s narrower interval says “met” while the honest t interval says the data cannot tell.

Worked case (the figure’s opening state). Five barley plots, t/ha, t/ha, contract “mean above 4.0 t/ha”. . Analyst Z: , entirely above 4.0. Analyst T: with , , which contains 4.0. Sign Analyst T’s report: the data cannot tell whether the contract is met.

The second dispute (Conclusion Builder) uses the same reasoning: choose the analyst whose method matches what is known about the spread, then place the contract value against that analyst’s interval.

Path B — The Report Architect

Task 1. The first error is one of: z used because of the sample size; ; ; or skipping the sampling and shape checks. Each draft carries its error forward consistently, so name the first wrong step.

Task 2. Decide the procedure from what is known about the spread: the sample’s → t with ; a calibration record’s → z. Then compute , the margin, and the bounds. The draft’s interval used the other critical value.

Revision note. Name the wrong choice (the critical value does not match what is known about the spread), give the corrected rule, and say what changes: the corrected interval carries exactly the uncertainty the data have, so its conclusion can be signed.

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.

Challenge 1 — Why

The values must total . Subtract the sum of the known values from to get the last one. Because the last value is forced once the mean is fixed, only values (equivalently, deviations) are free: .

Challenge 2 — How Quickly Does Converge to ?

At 95% confidence, for and for , so 28 is the first integer with a displayed value below 2.05. That landmark is only a closeness target; it does not make known or override the sampling and shape conditions.

Challenge 3 — Two Studies, One Evidence Question

Separate one-sample intervals are an indirect comparison. Non-overlap is strong but conservative evidence of a difference; overlap does not establish equality and can occur even when a direct comparison detects a difference. Use a confidence interval or test for to quantify the comparison directly.