L-INF-2: Solutions — Confidence Intervals for a Population Mean (Large Sample)

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.

The generated questions use new values on each attempt. Apply the workflow below to the values displayed in your version; the feedback attached to the problem gives the matching numerical answer.

Problem 1 — Constructing a Confidence Interval

For a sample with known population standard deviation:

1. Match the confidence level to its critical value:

   • 90%:
   • 95%:
   • 99%:

   A 95% interval leaves 2.5% in each tail, so . The z with is 1.645, the 90% value.

2. Compute the standard error.

3. Compute the margin of error.

4. Construct the interval.

Using itself as makes the interval too wide. Dividing by instead of makes it too narrow.

Problem 2 — May the Report Print This?

Sort each sentence by asking what it is about:

• Fine to print: the method captures the true mean at the stated long-run rate; “we are 95% confident the mean lies between the endpoints”; the endpoints bound the plausible values for the mean; this particular interval either contains or not.
• About individual customers: any sentence that puts a percentage of customers, or one customer’s wait, inside the interval. The interval estimates the mean; individual waits spread far wider.
• A probability on fixed : “a 95% probability that the true mean is between…”, “the true mean will fall inside 95% of the time”. The true mean does not move; the interval does.
• About other samples’ means: “a repeat study’s mean has a 95% chance to land inside”, “95% of all sample means lie between…”. Sample means cluster around , not around this interval’s centre, so a repeat study lands inside at most 95% of the time, and usually less.

Problem 3 — Determining Required Sample Size

Use the requested confidence level, known or planning value of , and target margin :

Then take the ceiling:

Evaluate the full ratio before squaring. Always round up; rounding down can exceed the allowed margin of error.

Problem 4 — Precision vs. Confidence Trade-off

The table uses one sample, so and are shared by all three rows. Only changes:

The 99% margin is always times the 90% margin. Multiplying by instead of is the most common slip.

For the colleague’s claim: neither confidence level is automatically best. Higher confidence misses less often but gives a wider, less informative interval. Choose the level before examining the data, based on the decision’s acceptable miss rate and the usefulness of the resulting precision.

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 — Can This Lesson’s Interval Be Used?

Check three things, in any order:

1. Sampling: is it an independent random sample of the population in question? A single class, a single pallet, or volunteers are not, and more data from the same source does not fix that.
2. Shape and size: is the population close to normal, or is the sample large enough for its skew? Strong skew with extreme values and a small fails.
3. : is the population standard deviation known? If only the sample’s is available, use the t procedure from L-INF-3, however large is.

Problem 2 — Constructing a Confidence Interval

The generated response is a pair of endpoints. The sample size is usually not a perfect square, so keep full precision while computing:

Round and only at the end, to two decimals. The interval is .

Problem 3 — One Commuter, or the Average Commuter?

The middle 95% of individual commutes reaches min from the centre. A confidence interval for the mean of 64 commuters reaches min. The same σ and the same 95% give very different ranges because one describes single values and the other describes a mean.

Problem 4 — Does the Mean Clear the Threshold?

Compare the threshold with both endpoints:

• Threshold below the whole interval: every plausible mean is above it, so the data show the mean exceeds it.
• Threshold above the whole interval: every plausible mean is below it, so the data show the mean is below it.
• Threshold inside the interval: plausible means lie on both sides, so the data cannot settle the question at this confidence level.

The confidence level describes how often the method captures the true mean across repeated samples, not a probability for this one interval, not a share of individuals, and not where a repeat study’s mean will land.

Problem 5 — Sample Size and the Quadruple Rule

For two target margins under the same confidence level and population spread:

Before ceiling rounding, when ,

The displayed integer requirements may not have an exact ratio of four because each raw result is rounded up separately.

Problem 6 — Diagnose Flawed Interval Work

Locate the first definite error, not merely a later line that inherits a wrong input. Depending on the generated version, the repair may be:

• use the that leaves half of in each tail (for 95%, 1.96, not 1.645);
• replace with for the standard error;
• divide by , not ;
• describe long-run procedure coverage rather than a probability for fixed ;
• round a required sample size up, never down.

Review Feed

For width questions, start from

A smaller , smaller , or larger makes the interval narrower.

For this lesson’s known- procedure, use

If only is supplied, the unknown- t procedure from L-INF-3 is the appropriate method.

Problem 7 — Full Synthesis: Sodium Intake

First construct the generated interval:

Then compare the whole interval with the threshold, as in Problem 4. The threshold may fall below, inside, or above the interval, and the conclusion changes with it. This conclusion concerns the population mean, not every individual.

For the tighter follow-up margin:

Mixed Review — Normal Probability

The generated stem states that the sampling distribution of is approximately normal. Standardize the supplied trigger:

For a lower-tail trigger, report . When the problem supplies a positive table value and the standardized trigger is :

Mixed Review — Sampling Distribution

First compute

For an independent random sample, a normal population gives a normal sampling distribution for . For a nonnormal population, a sufficiently large sample can support an approximately normal model when extreme skew, heavy tails, and influential outliers are absent; is a screening heuristic, not a guarantee. Standardize the threshold, then use the requested tail. For an upper tail:

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

Your explanation should include these ideas in ordinary language:

1. A sample mean is an estimate of an unknown population mean.

2. The margin of error is

   for this lesson’s known- model.

3. The reported interval is the sample mean plus and minus that margin.

4. The confidence level describes the long-run success rate of the interval-building procedure.

5. It does not mean that the fixed population mean has that probability of lying in this one computed interval, and it does not describe the percentage of individuals inside the endpoints.

You may formulate the answer on paper, in personal notes, or aloud before opening the model; OttoLearn does not need to collect or store the response.

Question 2 — Cold CI Construction

The generated scenario changes on each attempt and does not supply . Identify the method first: a large-sample confidence interval for a population mean with known . Then choose for the stated level and compute:

Keep full precision (the sample size is not a perfect square) and report to two decimals.

Question 3 — What May the Report Say?

The only printable sentence attaches the confidence level to the method: intervals built this way capture the true mean at that long-run rate, and this one either contains or not. The wrong options put a probability on fixed , describe individuals, or claim that a share of all sample means falls inside this particular interval.

Question 4 — Decide With the Interval

Compare the threshold with both endpoints, exactly as in Independent Practice Problem 4: below the interval, the mean exceeds it; above the interval, the mean is below it; inside, the data cannot settle the question.

Question 5 — Plan the Sample

Use the critical value for the stated level, evaluate the whole ratio, square it, and round up.

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 path. Each path has generated values, so the problem’s own post-attempt feedback supplies the numerical answer for your version.

Path A — The Replication Bet

Method choice. Your colleague lands inside your interval when their mean is within of yours: . Both means vary, independently, so the question is about the difference , whose mean is 0.

Spread of the difference. Variances of independent quantities add:

With equal sample sizes this is .

Standardize and look up.

Round to two decimals, then

For two studies of the same size at 95%, and the probability is about 0.835, well below 0.95. A larger repeat study wobbles less, so its mean lands inside more often, but still less often than the confidence level.

What the 95% promises. About 95% of intervals built this way capture the fixed true mean . It is not the chance that a repeat study’s mean lands inside your interval.

Path B — The Architect

Method choice. The decision concerns designing a study to meet a confidence and precision specification under a budget, so solve the sample-size formula before calculating cost.

Required sample size

Recruitment cost

Compare that cost with the stated cap: the cap may or may not cover it. If it does, the study can run as specified. If it does not, responsible alternatives include raising the budget, accepting a larger , or selecting a different confidence level when the decision context permits it. Do not round down or claim a precision the design cannot support.

Reflection. A strong response states whether the study is affordable, identifies which requirement it would renegotiate if not, explains the statistical consequence, and avoids declaring one confidence level universally correct.

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.

These three problems are optional stretch work beyond the lesson’s required outcomes. Their values regenerate.

Challenge 1 — Precision Has a Price

With confidence and fixed:

Halving the margin gives

Therefore,

The additional number of participants is

The additional funding is

Challenge 2 — Build a Critical Value from Scratch

For a central confidence level , write it as a decimal and compute

Use the supplied table excerpt to select the whose cumulative area is closest to that target; do not interpolate between rows. Retain full precision through the calculation, then round each endpoint to two decimal places with a final 5 rounded up. Then compute:

Challenge 3 — Plan with Imperfect Information

This problem supplies a plausible sensitivity range for . To protect the precision target throughout that stated range, use its upper bound for planning. This is conservative conditional on the supplied range; it is not a claim that the true spread is known.

At the monitoring cap:

Compare with the requested target:

• : the target is feasible under the stated approximation and cap.
• : the target is not feasible without more measurement windows, lower confidence when appropriate, a larger acceptable margin, or better planning information.