L-INF-5: Solutions — Hypothesis Testing for a Population Mean (σ Known)

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.

p-values below are written to four decimals. A p-value read from the printed z-table (with rounded to two decimals) can differ from the software value in the last digit; both are accepted, and the decision is never affected because every generated problem keeps clear of .

Workflow Check — What Must Come Before What?

Any order that settles the hypotheses and the conditions before the test statistic is correct. The p-value needs both the statistic and the direction of ; the conclusion comes last.

Problem 1 — Full z Test, Step by Step

This problem regenerates. For the displayed values:

1. Hypotheses. The question asks whether the mean has changed, so and .

2. Standard error.

3. Test statistic.

4. Two-tailed p-value. Read from the table, take the tail , then double it:

5. Decision. Reject if ; otherwise fail to reject. Never “accept”.

In this problem is a perfect square, so comes out exact.

Problem 2 — α, β, and Power

At a fixed sample size and effect, a lower raises and lowers power; a higher does the reverse. Increasing is the way to raise power without relaxing .

Problem 3 — Choosing the Test Form

Right-tailed: , . The planner asks only whether speed has increased. The form comes from the question, never from the sample.

Problem 4 — Classifying Errors

Type II error. (mean 14 days) was false (the true mean is 11 days), and the test failed to reject it.

Problem 5 — Full Test, Your Way

Same five steps as Problem 1, with the tail taken from the scenario’s claim:

• “has changed” → two-tailed, ;
• “increased above” → right-tailed, ;
• “fallen below” → left-tailed, .

In solo mode, type the p-value to four decimals; the steps then let you check your hypotheses and decision.

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.

Problems 1 and 2 — Full Tests with Untidy Numbers

The values regenerate. is not a perfect square, so keep the unrounded standard error:

Round to two decimals only to read the table. Choose the tail from the claim, then compare with the stated (0.01, 0.05 or 0.10).

When the sample falls on the “wrong” side. If but , then and

The data point away from the claim, so they are no evidence for it: fail to reject. Do not switch the direction after seeing the data.

Worked instance (two-tailed): , , , , .

From the table, , so ; software with the unrounded gives . Either way : reject . At the same data would not reject.

Problem 3 — Write the Conclusion

Three separate decisions:

1. Compare the printed with .
2. → reject ; → fail to reject . Never “accept”.
3. State the evidence in the words of : “sufficient evidence that the mean differs from / is above / is below ” when you reject, “not sufficient evidence that …” when you do not. Do not claim a direction on a two-sided test, do not claim proof, and do not claim the opposite direction when a one-sided test’s sample fell on the other side.

Problem 4 — Can This z-Test Be Used?

Run the z-test only when the sample is an independent random sample of the population in question, is known from outside the sample, and the population shape (or a large enough sample) supports a normal model for . Stop when only is reported (use the t-test of L-INF-6), when the observations come from one cluster, or when a small sample comes from a strongly skewed population.

Problem 5 — What Does This p-Value Tell Us?

• Correct reading: statements about what happens if is true (about 3% of no-effect trials would give a mean this low), comparisons with an fixed in advance, and the fact that varies from sample to sample.
• Turned around: “3% chance the drug has no effect”, “97% chance it works”, “3% chance the result happened by chance”, “the probability is true is 0.03”. A p-value gives no probability for either hypothesis.
• Overclaim: proof, a large effect, ” shows no effect”, and “a smaller p means a better drug”.

Problem 6 — Significant or Important?

Trial 1 (estimated benefit days, ): significant, and the estimate reaches the 1-day bar. Trial 2 (, benefit day, , , ): significant but far too small to matter. The sample size, not the effect, made Trial 2’s p-value smaller.

Problem 7 — Error Classification and the α–β Trade-Off

Anchor on what is true, then on what the test said: rejecting a true is Type I; failing to reject a false is Type II. At a fixed , lowering raises and lowers power.

Problem 8 — Failure to Reject Is Not Proof

Either the decision is reversed (”, so reject”), or a non-rejection is written as “accept ” or as proof. With , fail to reject: the evidence is insufficient, and a Type II error remains possible.

Problem 9 — Diagnose an Overclaim

Either “the p-value proves the mean is different” (a test gives evidence, not proof) or “statistically significant, so important in practice” (significance says nothing about size; judge the 0.2-unit difference in context).

Problem 10 — Low Power and Test Choice

With and unknown , the study may have low power, so a Type II error is possible, and the right procedure is a t-test, not this z-test.

Problem 11 — Multiple Comparisons

Running 20 tests at and reporting only the significant ones inflates the chance of at least one false positive: .

Mixed Review

Confidence-interval interpretation (L-INF-2). The 95% describes how often the method captures the fixed mean; inclusion leaves a value plausible but never pinpoints it.

Proportion interval (L-INF-4). Check and , then , keeping full precision until the endpoints.

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

A model answer is in the lesson. It must include: the p-value is computed assuming ; “at least as extreme” means in the direction(s) counts; and at least one overclaim to avoid (p as the probability that is true, p as an effect size, a large p as proof of ).

Question 2 — A Cold Test

Same method as §6 Problems 1–2, in new contexts. Read from the stem: it is 0.01 or 0.10 about half the time. Then complete the conclusion exactly as in §6 Problem 3.

Question 3 — What a p-Value Means

The flawed step reads as the probability that is true, or as the probability that is true. Neither probability is supplied by the test.

Question 4 — Errors and Power

A false alarm on a true is Type I; a missed real effect is Type II. Doubling at the same raises power; lowering at the same lowers it.

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 Regulator

File 1. The policy fixes from the side effects before the data: 0.01 (serious), 0.05 (mild), 0.10 (mild, no existing treatment). The claim is “shortens recovery”, so . With :

Worked instance: , gives , and (table: ). It rejects at 0.05 but not at 0.01.

File 2. Two separate questions. Is (any effect)? Is the estimated benefit at least 1 day (an effect that matters)? A very large trial can answer yes to the first and no to the second. Even when both are yes, the estimate is uncertain: the data are also consistent with smaller effects.

File 3. Count 1,000 drugs. With a share that work, power and level :

With , , : , not 5%.

🔍 The Auditor

The mandate fixes before sampling. Compute , , and , then compare with the office’s . About a third of the cases have below the target: then , , and there is no evidence of delay. The finding says so; it does not turn into a claim of faster service, which would need its own test planned in advance.

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 — The Critical-Value Approach

Find from the tail area puts in the rejection region:

• two-tailed: , giving , , for ;
• one-tailed: , giving , , .

Reject when falls in the region (, or ). This always agrees with comparing with .

Challenge 2 — Equivalence of a CI and a Two-Tailed Test

, keeping full precision through the margin. If lies outside the interval, the matched two-tailed test at rejects; if inside, it fails to reject.

Challenge 3 — Two-Sample Preview

Challenge 4 — How Large Must the Trial Be?

Power at a true mean below 14 is , so the target power needs :

For the drug (, , , power 0.80): , so patients. Always round up.