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

Module 4 · Statistical Inference

Section 1: Introduction

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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.

Imagine you run a drug regulator: a simplified, fictional agency that approves a drug when its trial passes a hypothesis test at the 5% level. Real agencies such as Health Canada never approve a drug on one trial, but the simplified version is enough to show something surprising.

A pharmaceutical company sends you its file. It claims its new drug brings mean recovery time below the industry standard of 14 days. Its trial of 49 patients produced a sample mean of 12.8 days. Hundreds of files like this one cross your desk every year, and about 5% of the time, a drug that does nothing will pass a test at the 5% level by luck alone.

Before you read on, make a prediction. Nothing is scored.

Predict first · not scored

Of the drugs that pass at α = 0.05, about how many actually do nothing?

“The sample average looks lower” is not enough to approve one drug. With only 49 patients, random variation could produce a mean of 12.8 days even if the drug does nothing at all. A hypothesis test asks a precise question: how unusual would our data be if the drug truly had no effect? You will settle this single trial in C4, and count out your inbox in C7.

After this lesson, you will be able to:

  • Choose a one-tailed or two-tailed alternative from the research question, then state and correctly.
  • Compute the z test statistic and its p-value when is known.
  • Say what a p-value means and what it does not mean, and make a decision in context without overstating the evidence.
  • Explain Type I and Type II errors, power, and why a statistically significant result need not be important.

Section 2: Prerequisites

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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.

What you need coming in, and why it matters today:

  • Sampling distribution of (L-INF-1, L-INF-2): varies from sample to sample with mean and standard error . Today’s test statistic is built directly on this idea.
  • Standard error formula (L-INF-2): . The denominator of the z test statistic is exactly this.
  • z-standardization (L-INF-2, L-DS-5): converts a sample mean to a z-score. We reuse this formula for a different purpose.
  • z-table and tail probabilities (L-PR-6): The standard normal table gives . Today you will use repeatedly to find p-values.
  • Sampling and shape conditions (L-INF-1): A random or otherwise independent sample, plus a population shape that makes the normal approximation reasonable, supports z inference. A large sample such as is a useful rule of thumb, not an automatic guarantee.

Quick check: can you recall these?

What is the formula for the standard error of the sample mean?

Bridge to today: In L-INF-2 you took the standardized value and built a symmetric interval around it. Here you take the same z-score and ask: “What is the probability of a value this extreme, or more, if is true?” That probability is the p-value. Same formula, a different question. This lesson keeps known; L-INF-6 (Hypothesis Tests for a Mean (σ Unknown) and for a Proportion) carries the same five steps to the t-distribution and to proportions.

Section 3: Core Concepts

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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.

How this section is organized: Nine concepts build on each other in order. The formula in C3 looks identical to L-INF-2’s standardization; what changes is what we do with the result.

  • C1–C2: The logic and structure of hypothesis testing
  • C3–C6: How to do it: the test statistic, the p-value (and what it means), two-tailed and one-tailed tests
  • C7–C9: What can go wrong: error types and your inbox, the α–β trade-off, significance versus importance, and “fail to reject”

C1 — The Logic of Hypothesis Testing

A hypothesis test starts from a default assumption, the null hypothesis, that nothing unusual is happening, and asks whether the data would be very unlikely if that assumption were true; very unlikely data count as evidence against it, just as a jury starts from “innocent” and convicts only on strong evidence.

In a criminal trial, the defendant is assumed innocent until the evidence proves otherwise. The jury does not have to prove innocence; the prosecution must prove guilt beyond a reasonable doubt. Hypothesis testing works the same way.

Null and Alternative Hypotheses

The null hypothesis is the default claim about the population, typically that a parameter equals a specific value: .

The alternative hypothesis is the claim we are trying to find evidence for: a departure from in some direction.

We write using ”=” only. We never “accept” ; we either reject it or fail to reject it.

Starting assumption
Criminal trial Presume innocence The defendant is innocent until proven otherwise.
Hypothesis test Assume H₀: μ = μ₀ We start by assuming the null hypothesis is true.
Weigh the evidence
Criminal trial Examine the evidence Is the evidence strong enough to overturn that assumption?
Hypothesis test Examine the data (p-value) How surprising is the data if H₀ were true?
The verdict
Criminal trial "Guilty" Evidence was strong enough. "Not guilty" Evidence fell short — not "innocent."
Hypothesis test "Reject H₀" Data was surprising enough. "Fail to reject H₀" Data fell short — never "accept H₀."
Absence of proof is not proof of innocence. A jury that says "not guilty" has not declared the defendant innocent — just as a test that "fails to reject H₀" has not shown H₀ is true. We never prove H₀; we only reject it or fail to reject it.
Figure 1: A criminal trial and a hypothesis test share the same logic. We assume a starting position (innocence ↔ H₀) and challenge it with evidence (testimony ↔ the p-value). Crucially, "fail to reject H₀" is the statistical "not guilty" — it never means we accept H₀ as true.

C2 — The Five-Step Framework

Every hypothesis test in this course, including L-INF-6’s, follows the same five steps; the details change but the steps do not.

The Five-Step Framework

  1. State and . Include the parameter (), the hypothesized value (), and the direction of (<, >, or ≠). Choose the significance level now, before the data.
  2. Check conditions. Verify independence, judge whether the population shape or sample size supports a normal approximation, and confirm that is known. If a condition fails, stop: this z-test does not apply.
  3. Compute the test statistic.
  4. Find the p-value. Use the z-table and the direction of to determine the tail area.
  5. State the conclusion in context. Compare p to : if , reject ; otherwise, fail to reject . Write the conclusion in a sentence about the real-world situation.

Step 5 conclusion language matters a great deal. Never write “we accept .” Never write “the data prove .” The only correct forms are “we reject ” (when ) and “we fail to reject ” (when ).

C3 — The Test Statistic

The test statistic measures how far the observed sample mean is from the hypothesized value, in standard error units. It is the z-score you computed in L-INF-2, but now , the null hypothesis value, is the reference point.

z Test Statistic for a Population Mean

where is the value stated in , is the known population standard deviation, and is the sample size.

A large means the data are far from what predicts: strong evidence against . A small means the data are consistent with .

Mini-example: Suppose , , , .

The sample mean is 3 standard errors above the null value. That is unusual, but whether it is unusual enough depends on the p-value.

A common error: using the sample standard deviation instead of the population . This lesson’s z-test requires a known . When is unknown, even for , the correct procedure is the t-test of L-INF-6. For large the two give nearly identical numbers, but the procedure is still a t-test.

Back to the drug trial: days, days known from years of records, patients. Figure 2 puts the sampling distribution of in a no-effect world above a ruler marked in standard errors. Before you drag: how many standard errors is 12.8 days from 14 days? Make your guess, then drag to 12.8.

Before you drag: how many standard errors is 12.8 days from 14 days?

One axis shown on two aligned scales: the sampling distribution of x̄ when H₀ is true, in days, on top, and the same line re-marked in standard errors below (1 SE = σ/√n = 0.6 days). x̄ starts at μ₀ = 14 days, zero standard errors away. Counting standard errors from μ₀ to an observed mean gives its test statistic z.
Drag x̄ (or use the arrow keys) and read the distance off the ruler.
Figure 2: The test statistic as a ruler reading. The blue curve is the sampling distribution of x̄ if H₀ is true (μ = 14 days), in days. The scale underneath is the same line, re-marked in standard errors (SE = σ/√n = 4.2/√49 = 0.6 days): that is the z-scale. Move x̄: the number of SEs it sits from μ₀ is z = (x̄ − μ₀)/SE. The axis arrows mark that the curve's tails continue past the drawn window.

With day, the trial’s mean sits : two standard errors below the standard.

C4 — The p-value

The p-value is the probability of a test statistic at least as extreme as ours, assuming is true; it is a share of no-effect worlds, not the probability that is true.

The company’s claim is directional (recovery below 14 days), so “at least as extreme” means a sample mean of 12.8 days or lower. Figure 3 opens empty. First commit to a reading of this trial’s p-value, then build the no-effect world and count.

This trial's p-value is about 0.023. Which statement is right?

Pick one reading to unlock the simulation.

Empty plot of the no-effect world, where the drug does nothing (μ = 14 days, SE = 0.6 days), with the observed mean x̄ = 12.8 days marked. Choose a reading of p ≈ 0.023, then simulate 500 trials to see what the p-value counts.

Figure 3: What a p-value counts. Each run simulates 500 trials of n = 49 patients in a world where the drug does nothing (μ = 14 days, σ = 4.2, SE = 0.6 days) and plots each trial's mean. Orange bars are the trials with a mean at or below the observed x̄: their share is the simulated left-tailed p-value, and the curve gives the theoretical one, Φ(z). The plot spans ±4 SE; any simulated means beyond it are reported as off-scale and still counted.

p-value

Decision rule: Reject if . Fail to reject if .

The significance level is set before the data are collected; it is never adjusted after seeing the result.

The single trial, settled. . If the drug did nothing, only about 2.3% of trials like this one would produce a mean of 12.8 days or lower. At that is rare enough: reject . The trial gives evidence that the drug shortens recovery.

That is not a 2.3% chance that the drug does nothing. Whether a drug that passes is a dud depends on something the p-value never sees: how many of the drugs in your inbox work at all. C7 counts it out.

The p-value is NOT the probability that is true. It is NOT the probability that the result occurred by chance. It is a conditional probability: given that is true, how likely is data this extreme? Confusing the two is the single most common error in applied statistics.

C5 — Two-Tailed Tests

When the alternative hypothesis is that the mean simply differs from , in either direction, we use a two-tailed test.

Two-Tailed Test

The p-value includes both tails:

We use because extreme evidence in either direction counts against .

Mini-example: , two-tailed test.

From the z-table, , so one tail is .

Both tails count: .

At : , so reject . At the same data would not be enough: .

A common error: computing only one tail area and forgetting to multiply by 2. For a two-tailed test, and are equally extreme, so both tails count.

C6 — One-Tailed Tests

When the research question is directional, asking specifically whether the mean is greater than or less than , we use a one-tailed test, and the tail is fixed by the question before the data arrive.

One-Tailed Tests

Right-tailed: , so

Left-tailed: , so

Mini-example (left-tailed): A call centre’s mean handling time has been 300 s. A manager claims new software has cut it: . The sample gives .

At , chosen before the data: , so reject .

The wrong side: if the sample had instead given (calls got longer), the same gives . The data point away from the claim, so they are no evidence for it at all.

The same z, three p-values. In Figure 4 the same test statistic is read against three alternatives, with the cut-off for your chosen drawn as a dashed line. Before you drag to : which panel rejects at ?

Before you drag to z = −1.73: which panel rejects H₀ at α = 0.05?

Significance level α
Two-tailed (Hₐ: μ ≠ μ₀)
p = 2·P(Z ≥ |z|)
Two-tailed (Hₐ: μ ≠ μ₀): the standard normal curve if H₀ is true, with the p-value area shaded and dashed critical values for α = 0.05.

Left-tailed (Hₐ: μ < μ₀)
p = P(Z ≤ z) = Φ(z)
Left-tailed (Hₐ: μ < μ₀): the standard normal curve if H₀ is true, with the p-value area shaded and dashed critical values for α = 0.05.

Right-tailed (Hₐ: μ > μ₀)
p = P(Z ≥ z) = 1 − Φ(z)
Right-tailed (Hₐ: μ > μ₀): the standard normal curve if H₀ is true, with the p-value area shaded and dashed critical values for α = 0.05.

Figure 4: The same z, three tests. Each panel shades its p-value area under the standard normal curve (the null world) and marks the critical value z* for the chosen α as a dashed line. Move z, or change α, and compare the plain-text decisions: the two-tailed p is always twice the smaller one-tailed p. Arrows at the panel edges mark tails that continue past the drawn window; every p-value includes the complete off-chart tail mass.

You must choose one-tailed or two-tailed from the research question, before collecting or examining the data. Choosing a one-tailed test after seeing which way the data point is called data snooping, and it inflates the true Type I error rate to roughly twice .

C7 — Type I and Type II Errors

Any decision rule can make two kinds of mistake: rejecting a true (a Type I error, a false alarm) or failing to reject a false (a Type II error, a missed effect).

Type I and Type II Errors

Type I error (false positive): Rejecting when is actually true. Its long-run rate is .

Type II error (false negative): Failing to reject when is actually false. Its rate is .

Statistical power: The probability of correctly rejecting when it is false. Power .

Students often confuse which error is which. A helpful mnemonic: Type I = false alarm (you cried wolf when there was no wolf). Type II = missed signal (the wolf was there and you said nothing). For your regulator, a Type I error approves a drug that does nothing; a Type II error turns away a drug that works.

Back to your inbox. Follow 1,000 candidate drugs through your test. Figure 5 counts them, and its last box answers the question from the Introduction. The share of drugs that really work (10%) and the power (0.80) are assumptions, not facts about real drug pipelines.

What happens to 1,000 drugs tested at α = 0.05 Natural-frequency tree. Of 1,000 candidate drugs, an assumed 10% (100) really work and 900 do nothing. With power 0.80, 80 of the working drugs pass and 20 are missed (Type II errors). With α = 0.05, 45 of the 900 duds pass by chance (Type I errors) and 855 are correctly stopped. 125 drugs pass, and 45 of them do nothing: 45 ÷ 125 = 36%, not 5%. α is the share of duds that pass, not the share of passing drugs that are duds.

125 drugs pass the test, and 45 of them do nothing: 45 ÷ 125 = 36%, not 5%.

Outcomes for 1,000 drugs tested
Drug Passes the test Does not pass
Really works (100) 80, correct 20, Type II error
Does nothing (900) 45, Type I error 855, correct
All drugs 125 pass; 45 ÷ 125 = 36% do nothing 875

Significance level α

If α drops from 0.05 to 0.01, will the 36% go up or down?

Predict first to unlock α.

α for every trial

Power

If power falls from 0.80 to 0.50, will the share of duds among approvals go up or down?

Predict first to unlock power.

Share of drugs that really work

If 30% of the candidate drugs worked instead of 10%, would the share of duds go up or down?

Predict first to unlock this share.
Figure 5: A regulator's 1,000 decisions, counted in drugs instead of probabilities. The share of candidate drugs that really work (10% to start), the power and α are assumptions to explore, not facts about real drug pipelines; every drug whose trial passes is approved. Counts are rounded to whole drugs and the percentage is worked out from the counts shown. Each control opens after you predict which way the share of duds among approved drugs will move. This way of counting false positives follows Colquhoun (2014).

Of the 125 drugs that pass, 45 do nothing: about 36%, not 5%. The 5% is the share of duds that pass; your question was the share of passes that are duds. Those are different fractions, and the second depends on how many candidate drugs work at all. That is why a p-value, on its own, can never tell you the probability that a drug works.

C8 — The α–β Trade-Off, and Significant ≠ Important

For a fixed sample size, decreasing (making it harder to reject ) increases (making it easier to miss a real effect); a larger sample is the main way to gain power without relaxing .

The α–β Trade-Off

For a fixed alternative, sample size, and analysis plan, lowering moves the rejection cut-off further into the tail and raises ; raising lowers but allows more Type I errors.

Increasing reduces the standard error and therefore lowers at a fixed . It does not change the preset .

Figure 6 puts the no-effect world (mean 14 days) beside a world where the drug really shortens recovery by 1 day (mean 13 days), for the trial’s . Its controls stay locked until you predict. If α drops from 0.05 to 0.01 with the same 49 patients, what happens to power?

Figure 6: The α–β trade-off in the drug trial. The solid curve shows where the sample mean x̄ lands if H₀ is true (μ = 14 days); the dashed curve shows where it lands if the drug really shortens recovery by one day (μ = 13). The test rejects H₀ when x̄ falls left of the cutoff 14 − z*·SE, with SE = 4.2/√n. A stricter α moves the cutoff left: fewer false alarms, but β grows and power falls. A larger n narrows both curves, so power rises at the same α. The shaded tails continue past both edges; the stated α, β and power include the complete off-chart tail mass.

At the trial has power of only about 0.51 against a 1-day effect, and about 0.25 at . It takes about 110 patients to reach power 0.80 at . There is no free lunch: decide which error costs more in your context, then design the study to make both acceptably rare.

Statistical Significance

A result is statistically significant at level if . This means only that the data would be unlikely under . It does not say the effect is large, important, or practically meaningful.

Which drug would you prescribe? Figure 7 shows two trials of two drugs, both significant at the 5% level. The one with the smaller p-value is the one with the smaller benefit.

A smaller p-value does not mean a bigger effect

Two trials, both significant: the smaller p-value has the tiny effect Two recovery-time trials against a standard of 14 days, σ = 4.2 days in both. Trial 1 has n = 49, x̄ = 12.8 days, an improvement of 1.2 days, z = −2.00 and one-tailed p ≈ 0.023; its x̄ ± 2 SE bar runs from 11.6 to 14.0 days, so the estimate is imprecise. Trial 2 has n = 10,000, x̄ = 13.9 days, an improvement of only 0.1 day (about 2.4 hours), z ≈ −2.38 and p ≈ 0.009; its bar runs from about 13.82 to 13.98 days. Trial 2 has the smaller p-value, but its improvement is far too small to matter: only Trial 1’s estimate is in the shaded region of a meaningful benefit, at least 1 day faster. A smaller p-value does not mean a bigger effect. Meaningful benefit at least 1 day faster (below 13 days) μ₀ = 14 days (standard) 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 Mean recovery time (days) Trial 1 · n = 49 x̄ = 12.8 days · z = −2.00 · p ≈ 0.023 x̄ ± 2 SE: 1.2 days faster, but imprecise Trial 2 · n = 10,000 x̄ = 13.9 days · z ≈ −2.38 · p ≈ 0.009 smaller p, tiny effect: 0.1 day (≈ 2.4 hours) Two trials, both significant: the smaller p-value has the tiny effect Two recovery-time trials against a standard of 14 days, σ = 4.2 days in both. Trial 1 has n = 49, x̄ = 12.8 days, an improvement of 1.2 days, z = −2.00 and one-tailed p ≈ 0.023; its x̄ ± 2 SE bar runs from 11.6 to 14.0 days, so the estimate is imprecise. Trial 2 has n = 10,000, x̄ = 13.9 days, an improvement of only 0.1 day (about 2.4 hours), z ≈ −2.38 and p ≈ 0.009; its bar runs from about 13.82 to 13.98 days. Trial 2 has the smaller p-value, but its improvement is far too small to matter: only Trial 1’s estimate is in the shaded region of a meaningful benefit, at least 1 day faster. A smaller p-value does not mean a bigger effect. Meaningful benefit: 1+ day faster (below 13) μ₀ = 14 days 11.0 12.0 13.0 14.0 Mean recovery time (days) Trial 1 · n = 49 x̄ = 12.8 · z = −2.00 · p ≈ 0.023 x̄ ± 2 SE: 1.2 days faster Trial 2 · n = 10,000 x̄ = 13.9 · z ≈ −2.38 · p ≈ 0.009 smaller p, tiny effect: 0.1 day (≈ 2.4 hours)
Figure 7: Two trials of recovery drugs against the standard μ₀ = 14 days, with σ = 4.2 days in both. Trial 1 (n = 49): SE = 4.2/√49 = 0.6, z = −2.00, one-tailed p ≈ 0.023. Trial 2 (n = 10,000): SE = 4.2/√10,000 = 0.042, z ≈ −2.38, p ≈ 0.009. Both reject H₀ at α = 0.05. Suppose a benefit only matters if recovery is at least 1 day faster (the shaded region): Trial 2 has the smaller p-value but an improvement of about 2.4 hours. Each bar is x̄ ± 2 SE, a rough guide to how precisely the trial pins down the mean.

Trial 2 has 10,000 patients, so its standard error is tiny ( day) and even a 0.1-day improvement, about 2.4 hours, sits 2.38 standard errors from 14. A p-value measures how surprising the data are under , not how big the effect is. Always report the estimated effect beside the p-value, and ask whether it is big enough to matter.

C9 — “Fail to Reject” ≠ “Accept”

To fail to reject means only that the data are not strong enough evidence against it at the chosen ; it does not show is true, because many other values remain just as plausible.

Fail to Reject

To fail to reject means the observed data do not provide sufficiently strong evidence against at the chosen .

It does not establish that is true. A real effect may be too small for the current study to detect.

When , we do NOT know that is true. This is the statistical equivalent of a jury returning “not guilty”: absence of proof is not proof of absence. Write “we fail to reject ” and nothing stronger.

A whole band of plausible values. A two-sided 95% z interval and the matching two-tailed z test at always reach the same decision: if falls outside the interval, the test rejects; if it is inside, it fails to reject. Drag in Figure 8 until 14 falls inside the interval, and look at what else the test would fail to reject.

A 95% z confidence interval for the mean recovery time and a two-tailed z test at α = 0.05 use the same full-precision critical value, so they always agree. When μ₀ = 14 lies outside the interval the test rejects H₀. When 14 lies inside, the test fails to reject, and it would fail to reject every other value in the interval too: failing to reject 14 does not show that 14 is the true mean.
  • 95% CI: x̄ ± z*·SE
  • μ₀ = 14, the value H₀ names
  • strip on the axis: x̄ values for which the test rejects μ = 14

Figure 8: A confidence interval and a two-tailed test give the same verdict. The bar is the 95% interval x̄ ± z*·SE for the mean recovery time (σ = 4.2, n = 49, SE = 0.6); the dashed line is μ₀ = 14. The interval and the test at α = 0.05 use z* ≈ 1.96 (the same full-precision value is used for both), so 14 outside the interval means the test rejects. When 14 is inside, the shaded band shows every hypothesised mean the test would also fail to reject: “fail to reject 14” is not “accept 14”.

Every value in the shaded band would survive the test, not just 14. That is exactly why failing to reject 14 cannot mean “the mean is 14”.

Why this wording matters downstream: L-INF-6, L-REG-3, and L-REG-4 all use the same five-step framework with the same conclusion language. Build the habit now.

Section 4: Worked Examples

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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.

Example 1 — Fully Worked: Cereal Fill Weight (Two-Tailed)

A cereal manufacturer claims its boxes contain g on average, with g. A quality inspector independently and randomly samples boxes and finds g. Historical fill-weight records show no severe skew or extreme outliers, so the normal approximation for is reasonable. Test at .

Step 1: State and

g     g (two-tailed: the inspector has no prior reason to expect low or high fill)

Step 2: Check Conditions

The sampling design provides independent observations, the historical shape evidence supports the normal approximation, and g is known. We may use the z-distribution.

Step 3: Compute the Test Statistic

I notice g.

The sign is negative because the sample mean fell below the null value.

Step 4: Find the p-value

Because is two-tailed, I use .

From the z-table: , so one tail is .

Both tails count: .

Software, which does not round to four decimals first, gives . The two differ only in the last digit, and both are accepted throughout this lesson.

Step 5: Conclusion

. We reject .

There is sufficient evidence at the 5% significance level to conclude that the mean fill weight is not 500 g. (Had the inspector chosen in advance, the same data would not have been enough.)

Example 2 — Predict the Decision: Student Commute Times

A CEGEP registrar independently and randomly samples students to test whether mean student commute time differs from 45 minutes. The population standard deviation is known to be minutes. The sample of students gives minutes. Historical commute-time records show no severe skew or extreme outliers.

The setup is versus , with . The sampling design provides independent observations, the historical shape evidence makes the planned normal approximation reasonable, and is known.

Predict first. The sample mean is about 3 minutes above 45. Is that more or less than two standard errors? Will the two-tailed p-value fall below 0.05? Write your prediction and the conclusion sentence you expect, then compute.

Show Solution

Step 1: Hypotheses

minutes

minutes

Step 2: Conditions

The sampling design provides independent observations, the historical shape evidence makes the planned normal approximation reasonable, and minutes is known.

Step 3: Standard error and test statistic

Step 4: Two-tailed p-value

From the table, , so one tail is .

(Software gives .)

Step 5: Decision and conclusion

, so fail to reject .

There is insufficient evidence at the 5% significance level that the mean student commute time differs from 45 minutes. That does not show the mean is 45 minutes: a 3-minute difference is less than two standard errors, too small for this sample to detect reliably.

Example 3 — Complete the Test: Sleep Hours

A health agency claims Quebecers sleep less than the national mean of 7.0 hours. Historical data give hours and show no severe skew or extreme outliers in sleep duration. An independent random sample of Quebecers gives hours.

Carry out the complete test at . Your response must choose the hypotheses from the claim, check the conditions, compute the test statistic and p-value, make the decision, and write one contextual conclusion.

Predict first. Which tail does the agency’s claim point to? Then: would you expect this p-value to beat , or only the more lenient ? Write the full test on paper before opening the solution.

Show Solution

Step 1: Hypotheses

hours

hours

The agency’s claim is directional, so evidence in the lower tail is relevant.

Step 2: Conditions

The sampling design provides independent observations, the historical shape evidence makes the normal approximation reasonable, and hours is known.

Step 3: Standard error and test statistic

Step 4: Left-tailed p-value

Using the rounded table value gives ; either route supports the same decision.

Step 5: Decision and conclusion

, so fail to reject .

There is insufficient evidence at the 1% significance level to conclude that Quebecers sleep less than the national mean of 7.0 hours. (Note: at , we would reject. The threshold matters, which is why it is fixed before the data.)

Example 4 — Diagnose and Repair a Research Decision

Read the following analysis carefully. It contains errors.

Analysis to examine:

A researcher tests kg at . The true population mean is actually 12.4 kg. The researcher obtains and concludes: “We accept . The mean is 12 kg.”

The researcher also notes: “Since we accepted , there is no Type II error here.”

Identify every incorrect claim, classify the decision outcome using the truth state of , and write a corrected two-sentence conclusion. Then name one design change that would reduce the chance of this kind of miss without changing the preset .

Predict first. How many errors do you see: one, two, or three? Which of the two error types from C7 actually happened here? Write your diagnosis before opening the solution.

Show Solution

Error 1 — Language: “We accept ” is never correct. Write “we fail to reject ”: failing to reject is not proof that is true.

Error 2 — Classification: Here is false, but the researcher failed to reject it. That is a Type II error (), not “no error.”

Corrected conclusion: At , the study fails to reject because . The data do not provide sufficient evidence that the mean differs from 12 kg, but this result does not establish that the mean equals 12 kg.

Design repair: At the same , a larger sample reduces the standard error and lowers , making a real 0.4 kg shift easier to detect.

Section 5: Guided Practice

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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.

Workflow Check — What Must Come Before What? (C2)

Before carrying out a complete test, arrange its actions by dependency. Stating the hypotheses and checking the conditions may happen in either order; what matters is that both are settled before the calculation they justify.

Logical Sequence
Arrange the hypothesis-test workflow in a defensible order
Drag or use arrow controls to order the steps.
Your Sequence

Problem 1 — Full z Test, Step by Step (C2 + C3 + C4 + C5)

Start with forced guidance: choose the hypotheses, then commit your standard error, test statistic, p-value, and decision. Each attempt uses a fresh scenario.

Problem 2 — α, β, and Power (C7 + C8)

After the trade-off figure, commit to the trade-off at a fixed sample size.

Problem 3 — Choosing the Test Form (C5 + C6)

A traffic planner wants to know whether the mean speed on a highway section has increased beyond the posted limit of 110 km/h. She samples 50 vehicles.

Problem 4 — Classifying Errors (C7)

A drug company tests “mean recovery time = 14 days” at . The company fails to reject . Later, an independent study confirms the drug does reduce recovery time (true mean = 11 days).

Problem 5 — Full Test, Your Way (C2 + C3 + C4 + C5 + C6)

You have practised each piece of the test separately. Now run one complete test and choose how much support you want: work through it one committed step at a time, or solve the whole problem on paper and type its p-value. Either way your first answer counts, and the steps stay available afterwards to check your hypotheses and decision.

Section 6: Independent Practice

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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.

From here on the numbers are untidy: sample sizes are not perfect squares, so keep the unrounded standard error, and z can be any value in the table. The significance level changes too (0.01, 0.05 or 0.10, always chosen before the data), and in a one-sided test the sample does not always fall on the side the claim predicts.

Problem 1 — Two-Tailed Full Test (C2 + C3 + C4 + C5)

Problem 2 — One-Tailed Full Test (C2 + C3 + C4 + C6)

Use the claim, not the observed sample, to choose the test direction.

Problem 3 — Write the Conclusion (C4 + C9)

The test is done; the sentence is yours. Each blank is checked separately.

Build the Conclusion
Complete the conclusion
Choose a phrase for each blank so that the sentence states the correct conclusion.

Problem 4 — Can This z-Test Be Used? (C2)

Step 2 can stop a test before it starts. Decide before computing anything.

Problem 5 — What Does This p-Value Tell Us? (C4)

Sort into groups
What does this p-value tell us?

A trial tests H₀: μ = 14 days against Hₐ: μ < 14 days for a recovery drug and finds p = 0.03. Sort each statement.

Drag an item into its group, or select it and then choose the group. On a keyboard: select an item and press the group’s number (1–3); 0 sends it back to Unsorted. Each item also has its own “move” buttons.
Unsorted

Problem 6 — Significant or Important? (C8)

These two trials are deliberately alike. For each, decide what the regulator can say, then use the feedback to name the one fact that changes the reading.

Structural Difference
Two trials, both significant
Identify what key word or condition separates Case A from Case B, then select the method for each.
Scenario A
Trial 1: n = 49 patients, x̄ = 12.8 days against the 14-day standard (σ = 4.2 days), one-sided p ≈ 0.023 at α = 0.05. A benefit matters if it is at least 1 day.
Scenario B
Trial 2: n = 10,000 patients, x̄ = 13.9 days against the 14-day standard (σ = 4.2 days), one-sided p ≈ 0.009 at α = 0.05. A benefit matters if it is at least 1 day.

Problem 7 — Error Classification and the α–β Trade-Off (C7 + C8)

Classify the outcome before examining the consequence for power.

Problem 8 — Failure to Reject Is Not Proof (C9)

Find the first error in the reasoning, then use the feedback to state the evidence correctly.

Problem 9 — Diagnose an Overclaim (C8 + C9)

Read the generated report as a reviewer. Locate the first invalid reasoning step, then replace it with a conclusion that separates statistical evidence, certainty, and practical importance.

Problem 10 — Low Power and Test Choice (C3 + C7 + C8)

A study uses an independent random sample of observations from a population without severe skew or influential outliers, has an unknown population standard deviation, and fails to reject after testing a new teaching method. Which diagnosis is best?

Problem 11 — Multiple Comparisons (C7 + C8)

A researcher runs 20 independent tests at and reports only the significant results. What is the central statistical problem?

Mixed Review — Confidence Intervals

Review Problem 1 — Confidence-Interval Interpretation (L-INF-2)

Find the first error in the student’s confidence-interval interpretation.

Review Problem 2 — Proportion CI Conditions and Construction (L-INF-4)

Check the conditions, compute the interval, then interpret the claim. Choose solo when you are ready.

Section 7: Mastery Check

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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

In your own words, explain what a p-value tells you and what it does not tell you. Write as if teaching a classmate who missed this lesson. Include the null-world assumption, the meaning of “at least as extreme,” and one common overclaim to avoid.

Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.

Compare with a model answer

A p-value tells you how surprising your data are, assuming the null hypothesis is true. Specifically: if the null were true, what fraction of random samples would produce a test statistic at least as extreme as the one you got?

A small p-value (say, 0.02) means your data would be rare in a null-hypothesis world: evidence against .

What a p-value does NOT tell you:

  • It is NOT the probability that is true. (Of the drugs that pass at the 5% level, the share that do nothing depends on how many candidates work at all; with the assumptions in Figure 5 it was about 36%.)
  • It is NOT the probability that your result occurred “by chance.”
  • It is NOT a measure of the size or importance of an effect: a huge trial can make a trivial effect significant.
  • A large p-value does not prove is true, only that you lack evidence against it.

Question 2 — A Cold Test

A fresh study, a stated significance level, and nothing else announced. Start on your own; open the steps only after your first answer if you need to find a slip.

Then write the conclusion for a second study of the same kind.

Build the Conclusion
Complete the conclusion
Choose a phrase for each blank so that the sentence states the correct conclusion.

Question 3 — What a p-Value Means

Read the report as a reviewer and find its first invalid step.

Question 4 — Errors and Power

Self-Assessment

50% confident
Still confusedReady for the Boss Fight

Section 8: Boss Fight

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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 of two missions. Neither tells you which method to use. Each starts with your own complete attempt; open the steps only if you need to find where it went wrong. Lose, and you get a rematch with new numbers.

⚖️ The Regulator

Back to your inbox. Set your threshold before the data, decide on a drug, judge whether its benefit matters, and count how many of your approvals are duds.

🔍 The Auditor

Your mandate says where to look for delays. The sample does not always cooperate.

Section 9: Challenge Problems

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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.

Optional stretch. These go beyond the lesson objectives. Attempt them when you feel confident with the core material.

Challenge 1 — The Critical-Value Approach

Decide without a p-value: find the cut-off from the z-table, then compare.

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

Construct the interval first. The post-attempt solution connects inclusion of to the matching two-tailed test decision.

Challenge 3 — Two-Sample Preview

This optional first look uses two independent samples and keeps its learner-controlled explanation after the generated prompt.

Challenge 4 — How Large Must the Trial Be?

The 49-patient trial had power of only about 0.51 against a 1-day effect. Design one that would not miss it so often.

Section 10: Solutions Reference

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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.

Complete, step-by-step solutions for all problems in Sections 5–9 are available on the solutions page. Solutions include full five-step write-ups, z-critical values shown explicitly, and decision rule justifications.

View Full Solutions →

If you’re stuck: Re-read the relevant Core Concept in Section 3. For z-test problems, make sure you used (population standard deviation) in the denominator and divided by , not by alone. For critical value lookups, confirm whether you are running a one-tailed or two-tailed test, as this changes your threshold. The solutions page shows the reasoning behind every step, not just the final answer.

Quick-Reference Formulas

One-Sample z Test Statistic (mean, known):

Decision Rule (P-Value Approach):

Decision Rule (Critical-Value Approach):

  • Two-tailed test (): Reject if
  • Right-tailed test (): Reject if
  • Left-tailed test (): Reject if

Standard Critical Values ():

  • : (two-tailed) or (one-tailed)
  • : (two-tailed) or (one-tailed)
  • : (two-tailed) or (one-tailed)