L-INF-6: Solutions — Hypothesis Tests for a Mean (σ Unknown) and for a Proportion

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 in the lesson’s tables are rounded to three decimal places, and z-table areas to four. Values below written with are rounded; decisions are always made at full precision.

Problem 1 — A t-Test, Step by Step

This problem regenerates. For the displayed values:

1. Hypotheses and procedure. . follows the question: if it asks whether the mean differs, or if a direction was named before the data. Only the sample’s is given, so is estimated: one-sample t-test.

2. Degrees of freedom. .

3. Statistic.

4. Bound the p-value. In the row of the excerpt (not the row , which the excerpt also shows), find the two critical values that bracket . Read the two-tail (2) headings for with , the one-tail (1) headings for a one-sided .

5. Conclude. Reject if the whole bound lies below 0.05; otherwise fail to reject. State the conclusion in the direction of , and never “accept ”.

In this problem is 9, 16 or 25, so is a whole number.

Problem 2 — A Proportion Test, Step by Step

1. Hypotheses. , with from the question.

2. Conditions. Random and independent (stated), and both and , computed from . If either count is below 10, stop: no normal-model p-value may be reported, whatever the sample’s own counts are.

3. Statistic.

4. p-value. Right-tailed: . Left-tailed: . Two-tailed: . A z-table reading with rounded to two decimals is accepted.

5. Conclude in context, in the direction of .

Problem 3 — Which Tool?

Ask in order:

1. Is there a claimed value? “How long do patients wait, on average?” has none: it needs an estimate (a t-interval, L-INF-3), not a test.
2. Mean or yes/no count?
3. For a mean: is known from outside the sample? Records, a specification or a calibration give : z-test, at any sample size (the filler with 10 bottles). Only : t-test, at any sample size (the 45 study-hour records).
4. The licence. A small, strongly skewed sample (the 12 emergency-room bills) is not a test yet. For a proportion, check the counts with : 12 of 40 households against a claim of 15% gives , so stop, even though the sample’s counts (12 and 28) look fine.

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 — Is the Mean Off the Claim?

The values regenerate, is not a perfect square, and the excerpt shows the rows , , and .

• given by records: the statistic is ; read the row.
• Only : the statistic is ; read the row.

Then bracket or in that row and read the headings that match .

Worked instance. 14 bolts, mm, mm, , .

In the row, , so : fail to reject (software: ). Reading the row by mistake gives and the bound , which would wrongly reject: the df slip changes the decision.

Problem 2 — A Directional Proportion Test

The direction was fixed before the survey, so the p-value counts one tail only.

Worked instance. A transit authority claims 45% of commuters ride with a monthly pass; before the survey a reviewer suspected the share is lower. In a random sample of 200, 77 do: . and , both at least 10.

(the z-table gives 0.0322). Since , reject : there is evidence that the share of commuters with a monthly pass is lower than 45%.

When a null expected count is below 10, the solo answer is “stop”: there is no trustworthy p-value from the normal curve.

Problem 3 — Diagnose the Test Workflow

Three report families, each with three named errors:

• Mean reports: using z because “a standard deviation is reported” when it is the sample’s ; reading the row; “accepting” after a large p-value.
• Proportion reports: building the standard error from ; using one tail for with ; “accepting” .
• t-table reports: ; one-tail headings for a two-sided test; claiming a bound past a critical value did not reach.

Find the first step that breaks the rules; later steps that follow from it are not separate errors.

Problem 4 — Same Rate, Bigger Survey

For each row, and . For the watchdog version (, ):

Quadrupling halves the standard error, so doubles. The difference never changes; only its precision does. That is why a report gives the estimated difference, not only the p-value.

Problem 5 — What May the Report Say?

Four decisions: the decision follows against (“reject” or “fail to reject”, never “accept”); the strength is “sufficient” or “insufficient evidence”, never “proof”; the direction matches as planned before the data; and the last sentence says what the test cannot show (the size or importance of a difference after a rejection, or that the claim is true after a failure to reject).

Problem 6 — Same Survey, Two Questions

Case A (estimate the rate) has no claimed value, so its standard error comes from the data: . Case B (test a claim of 15%) assumes the claim, so its standard error comes from : . The 95% interval, about , contains 0.15, which matches the two-sided test’s “fail to reject” at 0.05.

Mixed Review

Review 1 (L-INF-3). Read in the 95% column at ; then with the standard error unrounded, and .

Review 2 (L-INF-5). is known: , , .

Review 3 (L-INF-3). Not trustworthy: with and a strongly right-skewed population, the t-test’s p-value may be far off. Options: collect a larger sample, transform the data, or use a method that does not assume normality.

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 strong explanation says: the denominator is the spread of if were true; says the proportion is , so that spread is ; using would assume the sample is right in order to judge whether it is surprising. An interval has no claimed value, which is why it uses .

Question 2 — Which Procedure?

No claimed value → estimate. Mean with known → z-test; mean with only → t-test; yes/no count with a claimed share → one-proportion z-test. Then the licence: a small, strongly skewed sample, or a null expected count below 10, means “no test yet”.

Question 3 — A Cold t-Test

Same method as Independent Practice Problem 1 with unknown: , the row, and the headings that match .

Question 4 — May the Proportion Test Run?

Yes only when both and are at least 10. Neither nor the sample’s counts and is the check.

Question 5 — Significant, or Important?

A tiny p-value from a huge study shows the effect is detectable, not that it matters: judge its size in context. A large estimate from a tiny study that misses significance is inconclusive, not proof of no effect.

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 — The Fact-Checker

1. Standard error from , never from : the analyst who used is out.
2. Tails from the plan. If the direction (“higher”) was filed before the survey, the honest test is one-tailed. If it was chosen after seeing , the honest test counts both tails.
3. Sign the analyst whose choices match both.

For the opening survey (18 of 80, claim 15%, direction chosen after the data): , , and the honest p-value is two-tailed, : fail to reject at 0.05. Analyst 1’s would be honest only with a plan filed in advance; Analyst 3’s uses the wrong standard error.

Path B — The Claims Desk

The claim never names its method. Each case is one of five structures:

• t-test: a mean, only the sample’s , shape fine. Bound the one-tailed p-value at .
• z-test for the mean: a mean, from records. Use the row or the z-table.
• One-proportion z-test: a yes/no count against a claimed share, both null expected counts at least 10.
• Stop: a small, strongly skewed sample, or a claimed share that predicts fewer than 10 successes or failures.
• Estimate: a question with no claimed value (“how long?”, “what share?”): an interval, not a test.

“The data contradict the claim” means rejecting in the direction that disputes the claim, at .

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 — Paired Data

Treat the after-minus-before differences as one sample: with , two-tailed bound from the t-table, conclusion about the mean change.

Challenge 2 — How Many Bottles to Catch a 5 mL Drift?

A sample size always rounds up: (with 125 the power is just under 80%).

Follow-up. An exact t calculation needs 128 bottles, because the t-test estimates : its critical value is a little larger than 1.960 and its statistic has heavier tails, so a slightly larger sample is needed to reach 80% power.