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Advanced Statistical Methods

Hypothesis Testing Procedures

Published: 2026-08-11

Level: postgraduate

Audience: Postgraduate students in Advanced Statistical Methods

Prerequisite Knowledge

This lecture builds on the following concepts from earlier lectures. If any feel unfamiliar, review the linked notes before proceeding.

Previously Covered in This Subject

The standard normal distribution and z-scores — covered in Lecture 2
The sampling distribution of the sample mean and the central limit theorem — covered in Lecture 2
Hypothesis testing motivation from a coin tossing game — covered in Lecture 3
Null and alternative hypotheses, and type 1 and type 2 errors — covered in Lecture 3
The z transformation for sample means — covered in Lecture 3
Z versus t: choosing the distribution by sample size — covered in Lecture 3

4.1 Why Hypothesis Testing Exists

Hypothesis testing exists because every decision situation begins with two conflicting statements, and we must decide between them using limited data. You never have the whole population in front of you — only a sample — yet you still have to say yes or no: Is this batch of bricks acceptable? Is this coin fair? Should this drug be approved? Hypothesis testing is the disciplined answer to that problem: it turns "I think the data point one way" into "the data speak strongly enough, at a chosen risk level, for one side."

Hook: A coin lands heads 8 times in 10 tosses. Is it fair? Your gut says "maybe biased," but a fair coin can produce 8 heads too — just rarely. The question is not whether the data could happen under fairness, but whether the evidence is strong enough to overturn the starting claim. That judgment — made with a controlled error rate — is the whole business of hypothesis testing.

Last session introduced this idea with three examples. The first was a coin tossing example: toss a coin and decide whether it is fair or biased. The second was the criminal trial example — a fundamental and famous setup that maps hypothesis testing onto a courtroom, where we start from a position and let evidence shift the decision. The third was a personal example about evaluating questions, which showed that we often judge between claims without complete information.

4.1.1 Two Conflicting Statements

Every test starts by naming the two sides. The null hypothesis, written $H_{0}$ , is the statement we start from and try to overturn with evidence. The alternative hypothesis, written $H_{A}$ , is the claim we suspect might be true — the claim to be detected or supported. The two statements are logical opposites, so the evidence pulls toward one of them. The name "null" hints at its role: it is the default claim that stays standing unless the data speak against it.

Intuition — the courtroom picture. Think of $H_{0}$ as "not guilty" and $H_{A}$ as "guilty." The trial starts from not guilty; the prosecution must bring evidence strong enough to overturn it; the defense does not have to prove innocence. The jury decides with limited information — a sample of the full story, not the whole truth — exactly like a statistician deciding from a sample. Where the analogy breaks: a court's "not guilty" verdict does not mean the accused is innocent, only that the evidence was not strong enough. The statistical counterpart is "fail to reject $H_{0}$ ," which does not mean $H_{0}$ is true — only that the data did not overturn it.

The practical rule that will recur through this session: whatever is to be detected or supported is always stated in the alternative hypothesis. The null is the negation of that claim, and the test always begins by assuming $H_{0}$ is true. You never assume your suspicion — you make the suspicion prove itself.

4.1.2 The Two Types of Error

A decision based on data can go wrong in two ways. A type one error — rejecting $H_{0}$ when it is actually true — is denoted by alpha, $α$ , and is also called the level of significance. A type two error — accepting $H_{0}$ when it is actually false — is denoted by the Greek letter beta, $β$ . These two errors are the price of deciding from samples: no test avoids both, and the whole procedure is built around controlling them.

The four possibilities fit a two-by-two table. The rows are the conclusion you make; the columns are the truth you cannot see:

	$H_{0}$ is true	$H_{A}$ is true
Accept $H_{0}$	Correct decision	Type two error ( $β$ )
Reject $H_{0}$	Type one error ( $α$ )	Correct decision

Read the table one row at a time. If you accept $H_{0}$ and it is true, you are right; if you accept $H_{0}$ and it is false, you missed the effect — type two error. If you reject $H_{0}$ and it is true, you cried wolf — type one error; if you reject $H_{0}$ and it is false, you are right. The Greek letters $α$ and $β$ are the probabilities of the two wrong cells. The level of significance $α$ is special because the experimenter sets it in advance: by choosing $α = 0.05$ , you are saying "I am willing to reject a true null hypothesis five times out of a hundred." A high cost for a type one error calls for a smaller $α$ ; a low cost allows a larger one. The quiz default taught in class: when the problem does not give $α$ , take $α = 0.05$ .

Scope — what $α$ and $β$ cover, and what they do not. The two error probabilities apply to the long-run behavior of the test, not to any single run. Choosing $α = 0.05$ does not mean this one decision is 95% likely to be right. Also, most tests used in this course control $α$ only; $β$ is usually not fixed. For that reason the careful wording in the textbook is "do not reject $H_{0}$ " rather than "accept $H_{0}$ " — accepting $H_{0}$ puts you at risk of a type two error whose size you have not measured. In the exam, "accept the null" and "fail to reject the null" are treated as the same conclusion, but in a research write-up the phrase "do not reject" is the safer one.

Beyond the scope caveats, students in every class make the same small mistakes with the error vocabulary. A few of the most common ones are listed below.

Pitfalls — the classic error mix-ups.

Swapping the two errors. Type one = rejecting a true $H_{0}$ (false alarm, controlled by $α$ ); type two = accepting a false $H_{0}$ (missed signal, $β$ ). Test yourself with the table above until the mapping is automatic.
Reading "fail to reject" as "proved true." The data merely did not overturn $H_{0}$ . Weak evidence and a true null look identical from the outside.
Treating $α$ as the probability that the verdict is wrong. $α$ is only the probability of the type one cell, not of both wrong cells together.

Visual picture of the two errors. Draw the normal curve for the sample statistic under $H_{0}$ . The rejection region sits in the tail with area $α$ . The type one error is landing in that tail when $H_{0}$ is true. The type two error is failing to land there when the true curve has shifted away — the two curves overlap, and the overlap area is $β$ . Shrink the rejection region to protect against type one errors and the overlap grows: lowering $α$ raises $β$ . That trade-off is why both Greek letters matter.

Real-world connection. Drug regulators face both errors on every approval. Approving a new drug that is not safe is a type one error; failing to approve a safe and effective drug is a type two error. The regulator sets $α$ small precisely because the false-alarm cell is the costly one, but the second error can never be eliminated — only traded against the first. The same two-way risk shows up in manufacturing quality checks, spam filtering, and medical screening.

Recap + bridge. Hypothesis testing starts from two conflicting statements — $H_{0}$ , the default claim, and $H_{A}$ , the claim to be detected — and admits from the start that a decision from a sample can be wrong in two ways: type one ( $α$ , the level of significance) and type two ( $β$ ). With these two statements and these two errors named, hypothesis testing becomes a procedure we can follow step by step. The next topic shows the first step of that procedure: how to write the two statements correctly.

4.2 Writing the Null and Alternative Hypotheses

Hook — who carries the burden of proof? Before any arithmetic, a hypothesis test needs its two statements written exactly right. The single question that settles the writing: what is the person running the test hoping to establish? The claim they want support for is the alternative; everything else falls into place as its negation. Get this pair wrong and the whole test — its tail, its table lookup, its verdict — is wrong with it.

The textbook calls the alternative the research hypothesis when it states what the researcher hopes to prove: a new fuel injection system raises mileage, a new bonus plan raises sales, a new drug lowers blood pressure. In each of those cases the hoped-for improvement is the alternative, and the null is "no better than the old way." The alternative can also state an assumption to be challenged — for example, that a manufacturer's label is wrong. Either way, one question is the key: which statement, if supported by the data, would justify the action the decision maker wants to take?

4.2.1 The Core Guidelines

Two rules govern how the pair is written. First, whatever is to be detected or supported is always stated in the alternative hypothesis $H_{A}$ . Second, the null hypothesis $H_{0}$ is the negation of that statement, and the equality sign always lives in the null hypothesis. In many applications it is easier to identify the alternative first and then write its negation as the null; in other situations the reverse order comes more naturally, but either way the two statements must say the same thing. The procedure ends by accepting $H_{0}$ — the null survives unless the evidence is strong enough to reject it.

The three legal shapes. For a claim about a population mean, with $μ_{0}$ (mu-nought) the hypothesized value, there are exactly three forms:

Lower-tailed: $H_{0} : μ \geq μ_{0}$ against $H_{A} : μ < μ_{0}$
Upper-tailed: $H_{0} : μ \leq μ_{0}$ against $H_{A} : μ > μ_{0}$
Two-tailed: $H_{0} : μ = μ_{0}$ against $H_{A} : μ \neq = μ_{0}$

Notice what all three share: the equality symbol — $\geq$ , $\leq$ , or $=$ — sits in the null. The alternative carries only the strict inequality. The professor's rule of thumb: in the one-sided forms, writing the null with the plain equality (for example $H_{0} : μ = 10$ instead of $μ \leq 10$ ) is acceptable, because the equality sign belongs to the null and the arithmetic of the test uses the equality case anyway.

Warning — where the mistakes hide.

Equality in the alternative. Writing $H_{A} : μ = 10$ breaks the form: the alternative must be a strict inequality, or the test has no rejection region to aim at.
Direction backwards. If the hope is a reduction, the alternative is $<$ , not $>$ . A reversed pair gives the mirror-image test and a conclusion about the wrong side of the claim.
Two statements that disagree. "The two logical statements" must be exact opposites; if $H_{0}$ says $μ \geq 54$ and $H_{A}$ says $μ > 54$ , the pair overlaps and the test is ill-defined.

The three situations below show the pattern. In each one, read the situation once, decide what is to be detected or supported, then write the alternative first and negate it.

4.2.2 Situation 1: Microwave Radiation

The maximum acceptable level for exposure to microwave radiation in the United States is an average of 10 microwatts per square centimeter. It is feared that a large television transmitter may be polluting the air nearby, pushing the level of microwave radiation above the safe limit.

Worked example — building the hypothesis pair from the situation.

Read the situation. The exposure limit is an average of 10 microwatts per square centimeter; the fear is that the transmitter pushes the level of radiation above the safe limit.
Name the parameter. The situation talks about an average exposure level, so the parameter is the population mean $μ$ (in microwatts per square centimeter), and the boundary value is $μ_{0} = 10$ .
Identify the claim to be detected. What the situation wants support for is "average exposure above the safe limit" — that is the alternative:

$H_{A} : μ > 10$

Negate for the null. The null is the logical opposite, with the equality sign in it:

$H_{0} : μ \leq 10$ Writing $H_{0} : μ = 10$ is also acceptable, since the equality sign belongs in the null.

Check. The two statements are exact opposites, the equality lives in $H_{0}$ , and the $>$ in $H_{A}$ points right — so this is a right-tailed test, the first example of the one-sided family.

Final pair: $H_{0} : μ \leq 10$ against $H_{A} : μ > 10$ (or $H_{0} : μ = 10$ against $H_{A} : μ > 10$ ).

Why this ordering? The word that decides is "feared": someone wants to detect pollution above the safe limit. That detection goal is the alternative. The null is the safe-status quo — the level stays within the limit — and it takes strong data to overturn it. If the data cannot reject $H_{0}$ , the transmitter is allowed to keep running.

Real-world connection. This is environmental safety monitoring: agencies that set exposure standards for radio-frequency radiation check whether a transmitter pushes a populated area past its regulated limit. The 10 microwatts per square centimeter figure is a boundary of the same kind used in exposure guidelines for broadcast and communications equipment, where a false alarm costs a station operator money but a missed exceedance costs public health.

4.2.3 Situation 2: Power Steering for Vans

Design engineers are working on a low-effort steering system for vans, modified to fit the needs of disabled drivers. The old system required a force of 54 ounces to turn the van's 15-inch diameter steering wheel. It is hoped that the new design will reduce the average force needed to turn the wheel. Since the hope is a reduction, the alternative claims a smaller mean:

$H_{A} : μ < 54$

with the null as its negation:

$H_{0} : μ \geq 54$

Again, $H_{0} : μ = 54$ is fine. Here $μ$ is the average force, in ounces, needed to turn the wheel, and $μ_{0} = 54$ is the old system's average. The inequality points left, so this is a left-tailed test.

Why this ordering? "It is hoped that the new design will reduce" is the research claim. The engineers want support for less than 54 ounces, so $H_{A} : μ < 54$ . The null — "no better than the old system" — protects the old design from being discarded on weak evidence. Only if the sample mean falls far enough below 54 will the new system earn adoption.

Real-world connection. This is product design testing for accessibility: ergonomic engineering for vehicles adapted to disabled drivers, where the required steering force is a measurable design target. A wrong verdict matters both ways — adopting a system that is not actually easier burdens the very users it was meant to help, while rejecting a genuinely easier system blocks an improvement.

4.2.4 Situation 3: Computer System Turnaround Time

A computer system currently has 10 terminals and uses a single printer. The average turnaround time for the system is 15 minutes. Ten new terminals and a second printer are added, and we want to determine whether the average turnaround time is affected — we do not know whether it will go up or down. When the direction is unknown, the alternative sweeps both sides:

$H_{A} : μ \neq = 15$

and the null is:

$H_{0} : μ = 15$

Here $μ$ is the population average turnaround time in minutes, and $μ_{0} = 15$ is the current system's average. The alternative covers "affected" in either direction, so the test will be two-sided.

Why this ordering? The phrase "we do not know whether it will go up or down" removes any direction. The second printer could speed things up or the extra terminals could slow the queue — the engineers are not committing. A two-sided alternative matches an open question: detect any change, up or down.

Real-world connection. This is IT performance monitoring: before-and-after comparison of a system upgrade (adding terminals and printers) to decide whether the change moved the behavior of the system. The same "did anything change at all?" framing appears in equipment upgrades, process changes, and any experiment without a directional prediction.

4.2.5 The Three Situations Side by Side

Situation	What is detected/supported	Alternative	Null	Test type
Microwave radiation	Average exposure above the 10 limit	$H_{A} : μ > 10$	$H_{0} : μ \leq 10$ (or $= 10$ )	Right-tailed (one-sided)
Power steering	Average force below 54 ounces	$H_{A} : μ < 54$	$H_{0} : μ \geq 54$ (or $= 54$ )	Left-tailed (one-sided)
Turnaround time	Average affected in either direction	$H_{A} : μ \neq = 15$	$H_{0} : μ = 15$	Two-tailed

The pattern to take away: the alternative is the claim being argued for; the null is its negation with the equality sign; and the inequality in the alternative silently decides which kind of test you will run. That last step — reading the test type off the alternative — is exactly the next topic.

Recap + bridge. To write a hypothesis pair: (1) state what is to be detected or supported as $H_{A}$ ; (2) write the null as its negation with the equality sign; (3) check that the two statements say the same thing. The three situations gave a right-tailed pair ( $> 10$ ), a left-tailed pair ( $< 54$ ), and a two-tailed pair ( $\neq = 15$ ). The inequality carried by the alternative now determines the structure of the test — one-tailed or two-tailed — which is where the session goes next.

4.3 One-Tailed and Two-Tailed Tests

Hook — where does the evidence have to land? Suppose a sample mean lands slightly above the claimed value. Is that evidence against the null, or just noise? The answer depends on which direction counts as evidence — and that direction was already fixed when you wrote the alternative. A "greater than" claim only counts evidence that overshoots; a "not equal" claim counts overshooting and undershooting. The alternative's inequality is the arrow that points at the tail of the curve where evidence lives.

The previous topic produced three hypothesis pairs: $H_{A} : μ > 10$ , $H_{A} : μ < 54$ , and $H_{A} : μ \neq = 15$ . These are not three arbitrary shapes — they are the entire menu. The inequality in the alternative is a complete specification of the test's structure, and reading it correctly is the bridge between writing hypotheses and running a test.

4.3.1 The Three Test Types

The inequality inside the alternative fixes the kind of test. Greater than points to the right side of the distribution, so it is a right-tailed test. Less than points to the left, so it is a left-tailed test. Both of these are one-sided tests, also called one-tailed tests. Not equal to sweeps both sides, so it is a two-sided test, also called a two-tailed test. These are the only three possibilities — right-tailed, left-tailed, or two-sided — and the nature of the alternative tells you which one you have.

The direction map. The alternative's inequality is a pointer:

Alternative	Pointer direction	Test type
$H_{A} : μ > μ_{0}$	Points right	Right-tailed (one-sided)
$H_{A} : μ < μ_{0}$	Points left	Left-tailed (one-sided)
$H_{A} : μ \neq = μ_{0}$	Points both ways	Two-tailed

There are only three kinds of test — right-tailed, left-tailed, or two-sided — and no other kind exists. The question "is this test one-sided or two-sided?" is answered by one word of the alternative: greater than or less than means one-sided, not equal means two-sided. The professor's check from class: in the three situations of the previous section, the microwave case gave greater than (right-tailed), the steering case gave less than (left-tailed), and the turnaround-time case gave not equal (two-sided).

4.3.2 Rejection and Acceptance Regions

For a right-tailed test we cut the normal curve into two parts: the rejection region is the right tail, whose area is exactly $α$ , and the acceptance region covers the rest of the curve. The rule: reject $H_{0}$ when the computed statistic lands at or beyond the critical value:

$z \geq z_{α}$

where $z$ is the test statistic computed from the sample and $z_{α}$ is the table value read from the normal distribution at the given level of significance $α$ . For a two-sided test, $α$ splits into two halves, one in each tail, and the critical values become $\pm z_{α /2}$ ; the worked example in the next section shows that in full.

Visual picture — the cut normal curve. Draw the standard normal curve with the horizontal axis labeled $z$ (number of standard errors from the mean) and the vertical axis labeled probability density (the height of the bell). The curve is symmetric about $z = 0$ . For a right-tailed test, draw a vertical line at $z_{α}$ : the slim shaded sliver to its right is the rejection region, and its area is $α$ . Every computed $z$ that lands in that sliver counts as evidence strong enough to reject. The huge middle expanse to the left of the line is the acceptance region — landing there leaves the null standing. For a two-sided test, draw the same sliver at $+ z_{α /2}$ and its mirror image at $- z_{α /2}$ ; each holds area $α /2$ , and together they total $α$ . The one-sentence takeaway: the critical value is the boundary of the evidence; the rejection region is the tail whose area is the risk you chose ( $α$ ).

Pitfalls — reading the regions wrong.

One-sided critical value in a two-sided test. A two-sided test at $α = 0.05$ uses $\pm 1.96$ , not $+ 1.65$ . Using $α$ in one tail when the test is two-sided shrinks the rejection region and changes the verdict.
Flipping the direction. For a left-tailed test the rejection region is the left tail: reject when $z \leq - z_{α}$ . Keeping the right-tailed rule and testing a "less than" claim tests the wrong side.
Mixing up $z_{α}$ and $z_{α /2}$ . The notation carries the tail area: $z_{0.025} = 1.96$ (tail area 0.025), while $z_{0.05} = 1.645$ (tail area 0.05). Using one where the other is meant silently halves or doubles the risk.

Real-world connection. The tail logic is the same everywhere a one-directional claim is tested: a carmaker checking that a new brake part wears longer than the old standard uses a left- or right-tailed test depending on the claim's direction, while a manufacturer checking that a filling machine pours exactly the labeled volume uses a two-sided test because both underfill and overfill are bad. Choosing the wrong tail is a real business error, not just a notation slip: it either rejects improvements that are actually working or approves changes that are not.

Recap + bridge. The alternative's inequality fixes the test: $>$ gives a right-tailed test, $<$ a left-tailed test, $\neq =$ a two-sided test — and no other kind exists. The rejection region is the tail holding area $α$ , the acceptance region is the rest, and the boundary is the critical value ( $z_{α}$ one-sided, $\pm z_{α /2}$ two-sided). The remaining question is mechanical: how is the computed $z$ that we compare against the critical value actually calculated? That is the z test statistic — the next topic.

4.4 The Z Test for a Population Mean

Hook — how far from the claim is "too far"? When the claim is about an average — the mean thermal conductivity of a brick, the mean filling weight of a can — the data arrive as a single sample mean $x ˉ$ . The question every mean test answers is the same: the sample mean will almost never equal the claimed $μ$ exactly, so how far away from $μ$ must $x ˉ$ sit before we treat the difference as real evidence instead of luck? The z test gives that distance a number: it measures the sample mean in units of its own spread.

When the claim is about a population mean and the population standard deviation is known, we test with the z test, built on the normal distribution. The z test is the first of the two workhorse procedures of this session (the t test of section 4.7 is its close cousin); both share the same five-step skeleton, so mastering the skeleton now pays off twice.

4.4.1 The Test Statistic and Its Justification

The test statistic is:

$z = \frac{x ˉ - μ}{σ / n}$

where $x ˉ$ is the sample mean, $μ$ is the population mean claimed under the null hypothesis, $σ$ is the population standard deviation, and $n$ is the sample size.

Why this shape? Build it step by step.

The starting model. The population variable is $X$ with mean $μ$ and standard deviation $σ$ . The sample mean is the average of $n$ draws:

$x ˉ = \frac{x _{1} + x _{2} + \dots + x _{n}}{n}$

The sampling distribution. If the population variable $X$ follows a normal distribution, or if $n$ is large enough, then the sample mean $x ˉ$ also follows a normal distribution (this is the central limit theorem — the same sampling distribution idea covered in the earlier session). The mean of $x ˉ$ is $μ$ , and its standard deviation — called the standard error — is $σ / n$ . The standard error shrinks as $n$ grows: averaging more values pulls the sample mean tighter around the true $μ$ .

Standardizing. A normal quantity is converted into a standard normal z value by subtracting its mean and dividing by its standard deviation. Applying that rule to $x ˉ$ :

$z = \frac{x ˉ - μ}{σ / n}$

So $z$ answers: the sample mean is how many standard errors away from the claimed mean? A z of +1.77 says the sample mean sits 1.77 standard errors above the null's claim; a z of −2 says it sits two standard errors below.

Symbol registry (all four): $x ˉ$ (x-bar) is the sample mean, the average of the observed data; $μ$ (mu) is the population mean claimed under $H_{0}$ ; $σ$ (sigma) is the population standard deviation, assumed known; $n$ is the sample size. The denominator $σ / n$ (sigma over root n) is the standard error of the mean — how much a sample mean wobbles from sample to sample.

Scope — when the z test applies and when it does not.

Normality or large n. The standardization is exact when the population is normal. When it is not normal, the method is an approximation that improves as $n$ grows; in practice $n \geq 30$ is treated as large enough for most populations. With small $n$ from a non-normal population, the z test can mislead.
Known $σ$ . The formula needs the population standard deviation, not the sample's. In most real problems $σ$ is unknown — that case is exactly what the t test (section 4.7) is for.
Random sample. The sampling distribution story assumes the $n$ values are a random, independent sample; a biased sample breaks the machinery silently, because no formula can repair bad data collection.

Visual picture. Draw the sampling distribution of $x ˉ$ : a bell curve centered at the claimed $μ$ , with spread $σ / n$ , horizontal axis in the original units (e.g., conductivity, ounces, minutes). Now re-label the same curve in z units: the center becomes 0, and each tick mark one standard error wide. The z statistic is just the location of the observed $x ˉ$ on that re-labeled axis. Everything else in the test — the critical value, the p-value, the tail areas — happens on this standardized scale, which is why one normal table serves every problem.

Real-world connection. The z test is the default whenever a long-established process supplies a trusted $σ$ : filling machines with known machine variability, assay methods with published measurement error, well-characterized manufacturing lines. Quality control engineers use it to check whether a batch's average has drifted from the process standard — the same arithmetic the cement brick example in the next section runs by hand.

4.4.2 The Five-Step Procedure

Any hypothesis test follows the same simple five-step procedure, with the steps always in this order:

The five steps, and what each one is for.

State the null and alternative hypotheses. Write the two logical statements in the form of section 4.2 — the claim to be detected in $H_{A}$ , its negation with the equality sign in $H_{0}$ . This step decides the shape of the test (one- or two-sided).
Choose the level of significance $α$ . $α$ is the risk of a type one error you accept. If $α$ is not given, five percent is the standard default — you are allowed to pick $α = 0.05$ when the problem is silent. This step fixes how much evidence counts as enough.
Choose the appropriate test statistic and compute its value from the sample. For a mean with known $σ$ , that statistic is $z = (x ˉ - μ) / (σ / n)$ . This step turns the raw data into one comparable number.
Find the critical value, or the p-value. Read $z_{α}$ (one-sided) or $\pm z_{α /2}$ (two-sided) from the normal table, or compute the tail probability of the observed z. This step builds the benchmark to compare against.
Compare and conclude: reject $H_{0}$ or fail to reject it. Apply the rejection rule from section 4.3 (critical value approach) or compare p with $α$ (p-value approach), then translate the verdict into plain language about the real-world claim.

Steps 1, 2, and 4 come from tables and decisions; step 3 is the only arithmetic-heavy one; step 5 is a single comparison. The next section's cement brick example is a complete walk-through of all five steps, so if any step feels abstract here, it becomes concrete there.

Pitfalls — the usual z-test mistakes.

Plugging in $s$ for $σ$ . The sample standard deviation $s$ is a larger step up in notation but a different formula path entirely: $s$ in the denominator sends the test to the t distribution, not the normal one. The z test demands the known $σ$ .
$n$ in the wrong place. The standard error is $σ / n$ — the root is on the sample size, in the denominator of the denominator. Putting $n$ in the numerator inflates the statistic and can flip the verdict.
Using the sample mean as $μ$ . The $μ$ in the formula is the claimed value under $H_{0}$ (0.340 in the brick example), never the observed average.
Skipping step 2 and guessing a critical value. Without an $α$ , there is no boundary; the default is $α = 0.05$ .

Exam note: if the problem does not state alpha, choose $α = 0.05$ — the professor's standing instruction, and a question type you can bank on.

Recap + bridge. The z statistic $z = (x ˉ - μ) / (σ / n)$ measures how many standard errors the sample mean sits from the claimed mean, and the five steps (hypotheses → $α$ → statistic → critical value or p-value → conclusion) turn that number into a decision. The next section runs the whole procedure on real numbers — the thermal conductivity of cement bricks — and shows exactly how the critical value approach reaches its verdict.

4.5 Worked Example: Thermal Conductivity of Cement Bricks

Hook — a 0.003 discrepancy. A brick manufacturer claims its bricks have thermal conductivity 0.340. The lab measures 35 bricks and finds an average of 0.343 — off by 0.003. Three thousandths of a unit looks tiny; is it just measurement noise, or is the claim wrong? The five-step procedure from the previous section exists precisely to answer this: it converts "looks tiny" into "is tiny relative to the sampling spread?" — and the answer here turns out to be surprising.

This example is the session's first complete run of the five-step procedure, and it is worked twice — once with the critical value approach (this section) and once with the p-value approach (next section). Both runs must end at the same verdict; that agreement is itself a check on your arithmetic.

4.5.1 Problem Setup and Hypotheses

We test, on the basis of 35 determinations and at the five percent level of significance, whether the thermal conductivity of a certain kind of cement brick is 0.340 as claimed. Information gathered in similar studies gives the variability of such determinations as $σ = 0.010$ , and the mean of the 35 determinations is $x ˉ = 0.343$ . Since we have no reason to expect the true value to fall above or below the claim, the alternative is two-sided:

$H_{0} : μ = 0.340, H_{A} : μ \neq = 0.340$

with $n = 35$ and $α = 0.05$ .

Step 1 and step 2 of the procedure, named. The claim to be checked is "conductivity is 0.340"; the open question is whether it could be anything else, so $H_{A}$ sweeps both sides and the test is two-sided. $α = 0.05$ is given, so the risk of a type one error — wrongly accusing a compliant manufacturer — is fixed at five percent. The parameter $μ$ is the true average thermal conductivity of the whole population of these bricks, in the problem's conductivity units; the sample statistic $x ˉ = 0.343$ is our only window onto it.

Visual picture of the setup. Draw the sampling distribution of $x ˉ$ : a bell curve centered at the claimed 0.340, with spread $σ / n = 0.010/ 35 \approx 0.00169$ . The observed sample mean 0.343 sits a little way up the right side of the bell. The question is whether that point is so far out on the flank that the center 0.340 looks implausible.

Real-world connection. This is quality control for building materials: thermal conductivity determines how well a brick insulates, and a manufacturer's published value is a promise to builders. Checking a batch's average conductivity against the claimed value protects both sides — buyers get the insulation they paid for, and the manufacturer gets an early warning if the kiln process has drifted.

4.5.2 The Critical Value Approach

Step 3 — compute the test statistic. Substitute $x ˉ = 0.343$ , $μ = 0.340$ , $σ = 0.010$ , and $n = 35$ into $z = (x ˉ - μ) / (σ / n)$ . The division is worth doing line by line, because this is the arithmetic students most often mistrust:

$35 \frac{σ}{n} z = 5.916 = \frac{0.010}{5.916} = 0.00169 = \frac{x ˉ - μ}{σ / n} = \frac{0.343 - 0.340}{0.00169} = \frac{0.003}{0.00169} \approx 1.77 (square root of the sample size) (standard error, the denominator) (ratio of difference to standard error)$

The value $z \approx 1.7748$ rounds to z = 1.77 at two decimal places.

Step 4 — find the critical values. For a two-sided test at $α = 0.05$ , each tail holds $α /2 = 0.025$ , and the normal table gives:

$z_{α /2} = z_{0.025} = 1.96$

so the two critical limits are +1.96 and −1.96.

Step 5 — compare and conclude. The rejection rule for a two-sided test is to reject $H_{0}$ if $z \geq z_{α /2}$ or if $z \leq - z_{α /2}$ . Our computed value is $z = 1.77$ . Is $1.77 \geq 1.96$ ? It is not, so the computed value sits below the upper critical limit, inside the acceptance region, and we fail to reject the null hypothesis. The data do not overturn the claimed conductivity of 0.340 at the five percent level.

Worked example — the full run, one step at a time.

Hypotheses: $H_{0} : μ = 0.340$ , $H_{A} : μ \neq = 0.340$ (two-sided, because no direction is suspected).
Level of significance: $α = 0.05$ (given).
Test statistic: $z = (0.343 - 0.340) / (0.010/ 35) = 0.003/0.00169 = 1.7748$ , rounded to $z = 1.77$ .
Critical values: $\pm z_{0.025} = \pm 1.96$ .
Rejection rule: reject $H_{0}$ if $z \geq 1.96$ or $z \leq - 1.96$ .
Verdict: $1.77 < 1.96$ , so $z$ falls between the two limits, in the acceptance region: fail to reject $H_{0}$ .

Sense-check: the observed average 0.343 is only about 1.77 standard errors from the claimed 0.340, while the test demands about 1.96 standard errors of separation for a two-sided verdict at the five percent level. The evidence is suggestive but not strong enough — a plausible reading of "the claim stands."

4.5.3 Student Questions and Answers

Q: I doubted whether the computed z value of 1.77 is correct — how do we verify the calculation? A: Substitute $x ˉ = 0.343$ , $μ = 0.340$ , $σ = 0.010$ , and $n = 35$ into the formula $z = (x ˉ - μ) / (σ / n)$ ; the expression is the same one used throughout, and the value rounds to 1.77. Once the computation checks out, the comparison with 1.96 settles the conclusion — and if the comparison is hard to trust, redo the arithmetic: the two numbers are all that decide the verdict.

A second student probed the denominator itself, which is where the same arithmetic doubt keeps recurring.

Q: Are you doing the square root of 35 in the denominator? A: Yes. The denominator is $σ / n$ , so $35$ appears in the calculation; the expression is the same as the formula we wrote. You can redo the arithmetic yourself to confirm the value — $0.010/5.916 = 0.00169$ , and $0.003$ divided by that is $1.77$ .

4.5.4 The Verdict and Its Meaning

Visual picture — where 1.77 sits. Draw the standard normal curve with the horizontal axis in z units. Mark −1.96 and +1.96; the two tails beyond them are the rejection regions. The observed $z = 1.77$ sits just left of the +1.96 line, inside the acceptance region — close to the boundary, but not across it. The one-sentence takeaway: a statistic can be far enough to be interesting and still not far enough to be conclusive.

Pitfalls in the two-sided verdict.

Rejecting on one side only. The two-sided rule is a pair of inequalities: $z \geq 1.96$ or $z \leq - 1.96$ . A positive $z$ is compared with the positive critical value only; a negative $z$ with the negative one.
Rounding too early. $z = 1.7748$ rounds to 1.77 — but if you round the standard error to 0.0017 and then divide, the result drifts. Carry the unrounded arithmetic and round the final statistic.
Reading "fail to reject" as "proved equal." The test did not prove the conductivity is 0.340; it said the sample is consistent with 0.340 at the five percent level. The true mean could still differ by a small amount the data cannot detect.

Real-world connection — what the verdict does next. For the manufacturer, "fail to reject" means no corrective action: the batch is released, the claim stands, and the process keeps running. For a regulatory or standards context, the same verdict means no complaint is lodged. Note the asymmetry the procedure enforces: a wrong rejection would have stopped production and damaged a compliant manufacturer's reputation, so the test demands strong evidence before crying foul — that is the type one error protection built into $α = 0.05$ .

Recap + bridge. The cement brick test ran all five steps with real numbers: two-sided hypotheses at $α = 0.05$ , $z = 1.77$ from $x ˉ = 0.343$ against $μ = 0.340$ with standard error $0.00169$ , critical limits $\pm 1.96$ , and a verdict of fail to reject. That was the critical value approach. The next section re-runs the same example through the second procedure — the p-value approach — and shows why it reaches the same conclusion.

4.6 The P-Value Approach

Hook — a verdict with a size attached. The critical value approach ends with "reject" or "fail to reject" — two words that say nothing about how strong the evidence was. A test statistic of $z = 1.77$ and one of $z = 5.00$ both end in a verdict, but the two situations are not equally convincing. The p-value puts a number on that strength: it reports how probable the evidence is, which is why modern research articles prefer it over a bare verdict.

The professor's framing: behind the screen, software packages calculate p-values and print them by default; by simply looking at the output you can decide. Understanding how the number is built — the arithmetic in this section — is what makes interpreting that output reliable.

4.6.1 What the P-Value Is

The p-value is the area in the tail of the distribution beyond the observed value of the test statistic. Its working definition: it is the probability, under the null hypothesis, of getting a test statistic at least as extreme as the one we observed. When the computed z is positive, the p-value is $P (Z \geq z)$ ; when z is negative, it is $P (Z \leq z)$ . Software packages report this number by default, and modern research articles prefer the p-value approach over a bare verdict, because it says how probable the evidence is rather than just naming a conclusion.

Why "at least as extreme"? If the sample produced $z = 1.77$ , then any future sample that produced $z = 2.5$ or $z = 4$ is even stronger evidence against the null — so the probability that counts is the whole tail beyond the observed value, not just the single point. The smaller that tail area, the rarer the observed result would be if $H_{0}$ were true, and the more the data speak against it. For this reason the p-value is also called the observed level of significance: it is the smallest $α$ at which the test would reject, so a decision maker can compare the printed number with their own chosen risk.

Intuition — the surprise test. Suppose a magician claims a deck is standard, and you draw the ace of spades ten times in a row. Under the "standard deck" story, that run is wildly unlikely. That tiny probability is exactly what a p-value measures: how surprising the observed data would be if the starting claim were true. A very small p-value says "the data are very unusual under $H_{0}$ " — so unusual that the claim starts to look doubtful. The smaller the surprise, the more the evidence points at the alternative.

4.6.2 Computing P-Values: The Cement Brick Case

Continue with $z = 1.77$ from the cement brick example. Since z is positive, the one-sided p-value is:

$P (Z \geq 1.77) = 1 - P (Z \leq 1.77) = 1 - 0.9616 = 0.0384$

where 0.9616 is read from the normal table at 1.77. The decision rule: when p is less than or equal to alpha we reject the null hypothesis; when p is greater than alpha we fail to reject it.

Because the alternative is $μ \neq = 0.340$ , the test is two-sided and the p-value doubles:

$p-value = 2 \times 0.0384 = 0.0768$

Here $0.0768 > 0.05 = α$ , so we fail to reject the null — the same conclusion the critical value approach gave.

Worked example — the brick p-value, step by step.

Test statistic (from section 4.5): $z = 1.77$ .
One-sided tail: $P (Z \geq 1.77) = 1 - P (Z \leq 1.77) = 1 - 0.9616 = 0.0384$ .
Two-sided alternative $μ \neq = 0.340$ , so double: $p = 2 \times 0.0384 = 0.0768$ .
Compare with $α = 0.05$ : $0.0768 > 0.05$ , so p is greater than alpha: fail to reject $H_{0}$ .

Sense-check: the p-value approach and the critical value approach must agree — $z = 1.77$ falls short of the critical value 1.96, and the p-value sits above $α$ . Both procedures say the same thing: the evidence does not overturn the claim.

Careful with the direction of the rule. The comparison direction is the classic trap. p less than or equal to alpha means reject the null; p greater than alpha means fail to reject it. Small p-value, strong evidence, reject; large p-value, weak evidence, keep the null. Memorize it with the tail picture: a p-value is a tail area, so a small p-value means the observed statistic sits far out in the tail — far from the claimed center — which is exactly the situation that should reject the claim.

Why double? Alpha splits into two halves, $α /2$ on each side, and two times $α /2$ recovers $α$ ; the two-sided p-value doubles the one-sided value the same way. Because the alternative $μ \neq = 0.340$ counts extreme results on both sides, the evidence against the null is the combined area of both tails. The professor's equivalence, worth writing down: either compare the one-sided p-value with the one-sided alpha ( $0.0384$ vs $0.025$ ) or compare the doubled p-value with the full alpha ( $0.0768$ vs $0.05$ ) — both comparisons answer the same two-sided question.

A quick sanity check from the picture. The area to the right of 1.96 is 0.025, and 1.77 sits to the left of 1.96, so the area beyond 1.77 must be larger than 0.025 — which already signals p above alpha. The professor's rule of thumb from class: if z had come out near 3, the tail area would fall below 0.025 and the verdict would flip to rejecting the null. The tail picture lets you guess the verdict before touching the table: a statistic that clears its critical value has a p-value below $α$ ; one that falls short has a p-value above it.

4.6.3 Worked Cases: z = 2.10 and z = -1.75

Case two: the alternative is two-sided and the computed statistic is $z = 2.10$ . The p-value is:

$p-value = 2 \times P (Z \geq 2.10) = 2 \times (1 - P (Z \leq 2.10)) = 2 \times (1 - 0.9821) = 2 \times 0.0179 = 0.0358$

At $α = 0.05$ , p is less than alpha, so we reject the null hypothesis.

Worked example — z = 2.10, the rejecting case.

Alternative is two-sided (given), so double the upper tail.
Table read: $P (Z \leq 2.10) = 0.9821$ , giving tail $1 - 0.9821 = 0.0179$ .
$p = 2 \times 0.0179 = 0.0358$ .
Compare: $0.0358 < 0.05 = α$ , p is less than alpha: reject $H_{0}$ .

Sense-check: $z = 2.10$ clears the two-sided critical value 1.96, so it must also beat the p-value test — and it does.

Case three: the alternative is two-sided and the computed statistic is $z = - 1.75$ . The statistic sits on the negative side, so the one-sided p-value is the lower tail:

$p-value = 2 \times P (Z \leq - 1.75) = 2 \times 0.0401 = 0.0802$

Here p is greater than alpha, so we accept the null hypothesis. (The table value 0.0401 is the lower tail at $z = - 1.75$ , the value used in class; by symmetry it equals the upper tail at $+ 1.75$ , so the arithmetic is identical whichever side you read it from.)

Worked example — z = -1.75, the accepting case.

Alternative is two-sided, so double the lower tail.
Table read: $P (Z \leq - 1.75) = 0.0401$ .
$p = 2 \times 0.0401 = 0.0802$ .
Compare: $0.0802 > 0.05 = α$ , p is greater than alpha: fail to reject $H_{0}$ (accept the null).

Sense-check: $∣ z ∣ = 1.75$ falls short of the critical value 1.96, and the p-value $0.0802$ falls above $α$ — the two approaches again agree.

A quick exercise used in class to lock this in: three pairs of (p-value, alpha) values were shown; in two of them p exceeds alpha and the verdict is accept the null, and in one p is below alpha and the verdict is reject the null. The lesson is mechanical by design: compute the tail, double it if the alternative is two-sided, compare with alpha, and the verdict is forced — no judgment calls at the decision step itself.

4.6.4 Guidelines for Interpreting P-Values

Statisticians offer a ladder for describing p-values. The rung stated in class: a p-value below 0.01 is overwhelming evidence to conclude that the alternative $H_{A}$ is true. The textbook completes the ladder with three more rungs, and the full set is worth learning as a vocabulary for conclusions:

p-value range	Wording statisticians suggest
below 0.01	Overwhelming evidence to conclude $H_{A}$ is true
0.01 to 0.05	Strong evidence to conclude $H_{A}$ is true
0.05 to 0.10	Weak evidence to conclude $H_{A}$ is true
above 0.10	Insufficient evidence to conclude $H_{A}$ is true

When we reject the null we are accepting the alternative as true, and the strength of the language used in the conclusion depends on the context and on how small the p-value is. Note what the ladder does not say: none of the rungs assign a probability to $H_{A}$ itself — the p-value measures the data's surprise under $H_{0}$ , and the wording is a reporting convention, not a proof. In the exam, the rung that matters most is the one stated in class: a p-value below 0.01 lets you write "overwhelming evidence that the alternative is true."

Pitfalls — p-value abuse.

The p-value is not the probability that $H_{0}$ is true. It is the probability of the observed data (or more extreme) assuming $H_{0}$ . The two readings sound alike and are different.
Doubling a p-value in a one-sided test. Double only when the alternative is $\neq =$ . A one-sided alternative ( $>$ or $<$ ) uses the single tail.
Rounding the verdict instead of the p-value. The comparison is exact: $p \leq α$ rejects, $p > α$ fails to reject. A p-value of 0.0501 and one of 0.0499 land on opposite sides of a 0.05 line.

4.6.5 Student Questions and Answers

Q: I am still confused about how the p-value for z = 1.77 was calculated — can you explain it again? A: The p-value is the tail area beyond the observed statistic. For one side, $P (Z \geq 1.77) = 1 - 0.9616 = 0.0384$ . Because the alternative is two-sided, alpha divides into two halves, so the p-value doubles to $2 \times 0.0384 = 0.0768$ . Compare 0.0768 with alpha 0.05: it is larger, so we fail to reject the null hypothesis.

A second question tested whether the one-sided/two-sided choice was really at the student's discretion — it is not, and the answer explains why.

Q: How do we know whether a test is one-sided or two-sided? A: The nature of the alternative decides. Not equal means two-sided, because the value could be above or below the claim. Greater than or less than means one-sided. In the three situations from earlier: the microwave case gave greater than, a right-tailed test; the steering case gave less than, a left-tailed test; the turnaround-time case gave not equal, a two-sided test.

The last question of this topic was about the exam itself, and the professor's answer doubles as study advice.

Q: Will the exam ask us to calculate the z value and the p-value, or to interpret output? A: A few questions will ask you to work out the z value and the p-value. Most questions will give you an output (for example from Minitab or R) and ask you to extract the p-value, interpret the situation, and conclude. That is why both procedures matter: to interpret output confidently, you need to know how the printed p-value is built.

Recap + bridge. The p-value is the tail area beyond the observed statistic — $P (Z \geq z)$ for a positive z, doubled when the alternative is two-sided — and the verdict comes from comparing it with $α$ : $p \leq α$ rejects, $p > α$ fails to reject. The brick example gave 0.0768 (fail to reject), z = 2.10 gave 0.0358 (reject), z = -1.75 gave 0.0802 (fail to reject), always matching the critical value approach. The next topic asks what happens when the population standard deviation is unknown — the one-sample t test, where a new distribution takes over.

4.7 The One-Sample T Test

Hook — the common real-world case. The z test quietly assumed something most analysts do not have: a known population standard deviation $σ$ . In practice $σ$ is usually unknown — you have only the sample, and you must estimate both the center and the spread from the same handful of numbers. The t test is the z test's cousin built for exactly that situation, and the professor's line from class is the whole recipe: everything is the same except the formula, which uses $s$ instead of $σ$ .

Everything else in the procedure — the hypotheses, the level of significance, the five steps, the critical value and p-value comparisons — carries over unchanged. One new table (the t table) and one new number (the degrees of freedom) do all the real work.

4.7.1 Choosing Between Z and T

The z test needs the population standard deviation $σ$ , which we often do not have. The choice rule taught in class: when n is greater than 30 and sigma is known, use the z test; when n is small and sigma is unknown, use the t distribution. Everything else about the procedure stays the same.

The decision rule, as a two-case table.

Situation	Distribution	Test statistic
$n > 30$ and $σ$ known	Normal (z)	$z = \frac{x ˉ - μ}{σ / n}$
$n$ small and $σ$ unknown	t with $n - 1$ degrees of freedom	$t = \frac{x ˉ - μ}{s / n}$

The second row is the one added in this section. Note the pattern: the sample standard deviation $s$ replaces the population standard deviation $σ$ , and the letter on the test statistic changes from z to t.

Why does swapping $σ$ for $s$ change the distribution? When $σ$ is known, the denominator is a fixed number and the statistic is exactly standard normal. When $s$ is estimated from the same sample that produced $x ˉ$ , the denominator wobbles too — one sample might produce a slightly small $s$ , another a slightly large one. That extra wobble spreads the statistic out, so its curve is a t distribution: symmetric like the normal, but with heavier tails. The heavy tails are the price of estimating two unknowns from one sample. As the sample size grows, $s$ locks onto $σ$ , the t curve tightens, and for large $n$ the t and z curves become nearly indistinguishable — which is why the rule of thumb puts the boundary at about $n = 30$ .

Visual picture — t versus z. Draw both curves on the same axes (horizontal: test statistic value; vertical: density). The standard normal is a reference bell. The t curve is a slightly shorter, wider bell with fatter tails — more probability out where extreme values live. With $n - 1 = 4$ degrees of freedom the fatness is visible; with 30 degrees of freedom the two curves overlap almost completely. The one-sentence takeaway: the t distribution is the normal distribution with extra uncertainty added by estimating $σ$ .

Real-world connection. The t test is the everyday workhorse of small-sample studies: clinical experiments with a dozen patients, taste panels with a handful of tasters, factory trials with a few prototype runs. Whenever a researcher reports "I have data but no historical spread," the t distribution is the honest way to account for that ignorance — it refuses to let a small sample look more precise than it is.

4.7.2 The Sample Mean and Sample Standard Deviation

The t statistic:

$t = \frac{x ˉ - μ}{s / n}$

with $x ˉ$ the sample mean, $s$ the sample standard deviation, $μ$ the population mean under the null, and $n$ the sample size. The sample mean:

$x ˉ = \frac{\sum _{i = 1}^{n} x _{i}}{n}$

The sample standard deviation:

$s = \frac{\sum _{i = 1}^{n} ( x _{i} - x ˉ ) ^{2}}{n - 1}$

When a problem does not hand you $x ˉ$ and $s$ , compute them from the sample values with these two formulas.

Reading the two formulas. In $x ˉ = (\sum_{i = 1}^{n} x_{i}) / n$ , the symbol $\sum_{i = 1}^{n} x_{i}$ (the sum from i = 1 to n of the values) means "add up all $n$ observations," and the division by $n$ is the average. In $s = (\sum_{i = 1}^{n} (x_{i} - x ˉ)^{2}) / (n - 1)$ : first subtract the sample mean from each observation (each $x_{i} - x ˉ$ is a deviation from the average), square every deviation (so positive and negative deviations do not cancel), add the squared deviations, divide by $n - 1$ , and take the square root to return to the original units. The $n - 1$ (not $n$ ) is the degrees of freedom: once $x ˉ$ is known, only $n - 1$ of the deviations are free to vary, and dividing by the smaller number also keeps $s$ from understating the true spread of a small sample.

Pitfalls — computing or placing $s$ .

Dividing by $n$ instead of $n - 1$ in the standard deviation formula understates $s$ , and the mistake is silent — the number looks plausible. Always check: denominator $n - 1$ .
Forgetting the square root returns $s^{2}$ , the sample variance, which is not the statistic the t formula needs.
Mixing the two standard deviations: $σ$ (population, known) belongs to z; $s$ (sample, computed) belongs to t. Every exam mistake in this area traces back to that swap.

4.7.3 Worked Example: Glycerol in Wine

Glycerol is a major by-product of ethanol fermentation in wine production, and it contributes to the sweetness, body, and fullness of wines. A set of sample observations is available, and the desired average concentration value is 4. The question: does the sample data suggest that the true average concentration is something other than the desired value? "Something other than" means two-sided:

$H_{0} : μ = 4, H_{A} : μ \neq = 4$

with $μ$ the true average glycerol concentration. The test is a one-sample t test at the five percent level of significance. Five sample values were given ( $n = 5$ ), and the mean and standard deviation computed from them go into the formula. The computed t statistic is:

$t = - 0.58$

On the given numbers. The five sample observations were not read aloud in the lecture, so $x ˉ$ and $s$ are taken as given by the problem; the value $t = - 0.58$ is what those summary values produce through $t = (x ˉ - μ) / (s / n)$ . What the number says: the sample mean sits 0.58 standard errors below the desired value 4. And the number is internally consistent with the software output of the next subsection: for $t = - 0.58$ with 4 degrees of freedom, the two-sided p-value is 0.594 — exactly the printed value.

Degrees of freedom are $n - 1 = 4$ . At the five percent level the t table gives:

$t_{α /2} = t_{0.025} = 2.776$

and by symmetry the critical limits are $+ 2.776$ and $- 2.776$ , with area 0.025 in each tail. The rejection rule mirrors the z test: reject $H_{0}$ if $t \geq t_{α /2}$ or if $t \leq - t_{α /2}$ . Our value $t = - 0.58$ lies between the two limits, inside the acceptance region, so we accept the null hypothesis.

Worked example — the glycerol test, complete.

Hypotheses: $H_{0} : μ = 4$ , $H_{A} : μ \neq = 4$ (two-sided — "something other than").
$σ$ is unknown and $n = 5$ is small, so the t test applies.
Test statistic: $t = (x ˉ - 4) / (s / 5) = - 0.58$ (using the $x ˉ$ and $s$ computed from the five observations).
Degrees of freedom: $n - 1 = 4$ . Critical values at $α = 0.05$ : $\pm t_{0.025} = \pm 2.776$ , area 0.025 in each tail.
Rejection rule: reject $H_{0}$ if $t \geq 2.776$ or $t \leq - 2.776$ .
Verdict: $- 2.776 < - 0.58 < 2.776$ , so $t$ falls inside the acceptance region: accept $H_{0}$ (fail to reject).

Sense-check: a statistic only 0.58 standard errors from the claim is nowhere near the boundary at 2.776 standard errors — the sample is comfortably consistent with the desired concentration of 4.

Visual picture. Draw the t curve with 4 degrees of freedom, mark $\pm 2.776$ , and shade the two tails (each area 0.025). The observed $t = - 0.58$ sits near the center, far from either shaded tail — the picture makes the verdict obvious before any comparison of words.

Real-world connection. Glycerol concentration is an actual quality measure in wine production, where the compound adds sweetness, body, and fullness to the wine. A winery checking a production lot against its target concentration runs exactly this test; with only a few samples per lot, the t test — not the z test — is the honest instrument, because the batch-to-batch spread is not known in advance.

4.7.4 Reading the Minitab Output

Output from a software package such as Minitab gives everything at once: the mean is calculated, the standard deviation is given, and the t value is printed, with the p-value as the last number to interpret. The p-value approach works exactly the same for t as for z. In this run the p-value is:

$p-value = 0.594$

Since $0.594 > 0.05$ , p exceeds alpha; it also exceeds any other reasonable significance level such as 0.10, which strengthens the verdict: we accept the null hypothesis. The sample does not show the true average glycerol concentration differing from the desired value of 4.

Worked example — interpreting the output.

A printed one-sample t output shows, in order: $x ˉ =$ (the sample mean, computed by the software), $s =$ (the sample standard deviation), $t = - 0.58$ (the test statistic), and $p-value = 0.594$ (the last number to interpret). The steps: identify the test as two-sided from $H_{A} : μ \neq = 4$ , take the printed p-value 0.594, compare with $α = 0.05$ : $0.594 > 0.05$ , and also with $α = 0.10$ : $0.594 > 0.10$ . Verdict: accept $H_{0}$ — the p-value is larger than any reasonable significance level, so the evidence for a change is essentially nil.

The professor's exam note: the p-value is the last number on such an output and it is the number to extract. Whether the output comes from Minitab or from R (the free software package used to show outputs in this course), the reading procedure is the same: find the p-value, compare it with $α$ , conclude.

Real-world connection. Statistical software is how this test is run in practice — Minitab in industry, R and Python in research. The whole point of the earlier arithmetic is that it lets you check the printed numbers and, more importantly, know what the software's last line actually means.

4.7.5 Student Questions and Answers

Q: Does the p-value approach work the same way for the t test as for the z test? A: Yes — the approach is identical. The only difference is the formula, which uses s instead of sigma. In the output the p-value is 0.594, greater than alpha 0.05, so we fail to reject the null hypothesis, and both approaches give the same conclusion. The critical value comparison and the p-value comparison are two routes to the same verdict, for z and for t alike.

Recap + bridge. When $σ$ is unknown and $n$ is small, the t distribution replaces the normal: $t = (x ˉ - μ) / (s / n)$ with $n - 1$ degrees of freedom, everything else unchanged. The glycerol example accepted the null (t = -0.58 inside $\pm 2.776$ , p = 0.594), and the Minitab output supplied the p-value directly. The next topic closes the session by previewing what happens when the claim is about variance rather than the mean — a new distribution, the chi-square, enters the picture.

4.8 The Road Ahead: Variance and the Chi-Square Distribution

Hook — a tie that hides everything. Two drugs were compared, and both produced an average of 70 heartbeats per minute. By the averages, they are identical — a 70-70 tie that carries no information. Yet one drug is worse for patients in a way the average cannot show. How can the verdict be visible in the data and invisible in the average? Because the average is not the whole story: the spread is. This section closes the lecture by pointing at the parameter that captures spread — the variance — and at the new distribution that tests it.

Everything so far tested claims about a population mean $μ$ with the z and t distributions. The natural next question: what about claims about how spread out the population is? The logic of writing hypotheses carries over unchanged; only the distribution of the test statistic changes.

4.8.1 Why Variance Matters

A drug comparison example from the very first class showed why variance matters. The old drug and the new drug both produced an average of 70 heartbeats — a 70-70 tie that carried no information. But the data told a different story: the new drug produced extreme reactions, and that spread is captured by a simple distribution parameter, the variance. It was the variance that let us conclude the old drug behaves better than the new drug. The message: averages can hide differences that variance exposes.

Intuition — darts at a bullseye. Two players' darts both land, on average, in the bullseye — but one player's throws cluster tightly around it while the other's scatter across the board, with a few near-misses and a few wild shots. Same average, different players. The variance is the "how spread-out" number: it summarizes the scatter around the average. The patient analogy is exact: an average of 70 can come from 35 patients all near 70 (old drug) or from patients swinging between 50 and 90 with extreme reactions (new drug). Averages describe where the center sits; variance describes how far the tails reach — and for safety decisions, the tails are often what matter.

Real-world connection. This is pharmaceutical testing: comparing old and new drugs on patient response, where spread matters as much as the average. A drug whose average effect is fine but whose individual responses swing wildly is dangerous — the extreme reactions are exactly the side-effect cases a regulator worries about. The same logic governs manufacturing tolerances (a part line with correct average but wild variability fails quality checks), finance (two investments with equal average return but very different volatility), and any process where consistency is a property in its own right.

4.8.2 The Chi-Square Statistic

When the problem talks about variance or standard deviation rather than the mean, the same logic applies for writing hypotheses — $σ^{2} \neq = σ_{0}^{2}$ , $σ^{2} > σ_{0}^{2}$ , or $σ^{2} < σ_{0}^{2}$ — but a new distribution enters: the chi-square distribution. The statistic is:

$χ^{2} = \frac{( n - 1 ) S ^{2}}{σ ^{2}}$

where $S^{2}$ is the sample variance and $σ^{2}$ is the population variance.

Symbols and structure. The Greek letter $χ$ (chi, pronounced "kai") gives the distribution its name; $χ^{2}$ is the symbol for the statistic and for the distribution. $S^{2}$ (capital S squared) is the sample variance — the square of the sample standard deviation, $S^{2} = \sum_{i = 1}^{n} (x_{i} - x ˉ)^{2} / (n - 1)$ . $σ^{2}$ (sigma squared) is the population variance claimed under the null. $n$ is the sample size. The shape of the fraction: the sample's observed spread is compared with the claimed population spread, scaled by the degrees of freedom $n - 1$ . The null hypothesis is always stated in terms of $σ^{2}$ (or $σ$ ) — never $μ$ — with the equality in the null and the claim to be detected in the alternative, exactly as in section 4.2. A key fact from the reference treatment: when a random sample of size $n$ comes from a normal population, $(n - 1) S^{2} / σ^{2}$ follows a chi-square distribution with $n - 1$ degrees of freedom.

The fundamental difference the professor asked you to find. Compare $χ^{2} = (n - 1) S^{2} / σ^{2}$ with $z = (x ˉ - μ) / (σ / n)$ . The z statistic can be positive or negative — a sample mean can sit above or below the claim. The chi-square statistic cannot be negative: $S^{2}$ and $σ^{2}$ are both squared quantities, so their ratio is always greater than or equal to zero. That single fact changes the entire picture:

Visual picture — the chi-square shape. Draw the horizontal axis from 0 to the right, labeled $χ^{2}$ (possible values start at 0 and go up); the vertical axis is the density. Unlike the symmetric bell of z and t, the chi-square curve is skewed — it rises from zero, climbs to a peak, and falls back in a long right tail. With few degrees of freedom the curve is heavily skewed; as the degrees of freedom grow, the peak moves right and the curve becomes more bell-like, approaching a normal shape for large $n$ . Landmarks: the entire curve sits on non-negative values (no left tail exists), and the mean of the distribution is its degrees of freedom. The one-sentence takeaway: a test statistic that can only be positive needs a distribution that only has positive values — that is the chi-square.

Scope — variance tests and their new assumptions. The chi-square statistic is built for claims about $σ^{2}$ : two-sided ( $σ^{2} \neq = σ_{0}^{2}$ ), upper-tailed ( $σ^{2} > σ_{0}^{2}$ ), or lower-tailed ( $σ^{2} < σ_{0}^{2}$ ). The chi-square distribution of $(n - 1) S^{2} / σ^{2}$ is derived under the assumption that the sampled population is normal; departures from normality weaken the procedure, especially with small samples. Its tables are read differently from the z and t tables — one-sided by design, because the statistic has no negative side — and the rejection region always sits in the right tail (large $χ^{2}$ means the sample variance exceeds the claimed value by too much). The details of the tables and the decision rules are the business of the next session.

4.8.3 What Comes Next

Next session covers what the chi-square distribution is, how its tables are extracted, and three applications where it is used extensively; among many possible applications, three were pointed out. The three standard applications of the chi-square, which the reference material groups together, are: the goodness-of-fit test (does the data fit a claimed distribution?), the test of independence of variables (are two categorical variables related?), and the test of equality of more than two population proportions. These are the three applications to watch for.

Exam note: you can expect any one of these chi-square applications in the mid-semester exam — the professor flagged them as very interesting applications with real-time relevance, worth preparing specifically.

Practice problems with solutions will be shared in Teams files and Google Drive, and outputs will be shown with R, a free software package — the same "extract the p-value from the output" skill from the t-test section will apply there.

Recap + bridge. This session built the machinery of hypothesis testing: the two conflicting statements with their two error types (4.1), writing $H_{0}$ / $H_{A}$ with the equality in the null (4.2), the three test types fixed by the alternative (4.3), the z test and its five-step procedure for means with known $σ$ (4.4–4.5), the p-value approach that software prints (4.6), the t test for means with unknown $σ$ (4.7), and now the preview of variance testing (4.8): same hypothesis logic, new statistic $χ^{2} = (n - 1) S^{2} / σ^{2}$ , new distribution — asymmetric, non-negative, with its own tables and three applications. The next session reads those tables and runs those applications.

Exam Guidance Summary

Everything the professor said about the quiz and the mid-semester, in one place.

Quiz one runs from 16 August to 30 August with a single attempt; the window is wide, so timing is flexible. It draws on chapters six, seven, and eight; from chapter six only the normal distribution problems covered in class are included. Expect 25 elementary objective-type questions worth five marks each, at basic level.
Practice problems for the quiz and the mid-semester will be shared with solutions in Teams files and Google Drive; the textbook carries more practice problems.
Question pattern for tests: a few questions ask you to compute the z value and the p-value; most questions give you an output (for example from Minitab or R) and ask you to extract the p-value, interpret the situation, and conclude. To interpret output confidently, understand how the p-value is built — that is why both the critical value approach and the p-value approach are examined.
If the level of significance is not given, choose the five percent level. This is the standing default: $α = 0.05$ when the problem is silent.
Use the p-value rule: p less than or equal to alpha means reject the null; p greater than alpha means fail to reject it. Getting the direction of this rule backwards is the most common exam error.
Interpret p-values with the guideline ladder: a p-value below 0.01 is overwhelming evidence that the alternative is true; between 0.01 and 0.05 is strong evidence; between 0.05 and 0.10 is weak evidence; above 0.10 is insufficient evidence. A p-value above 0.05 — and above any other reasonable level such as 0.10 — means we accept the null hypothesis.
Chi-square applications: you can expect any one of them in the mid-semester exam, so prepare all three (goodness of fit, test of independence, equality of more than two proportions) when they are covered in the next session.
Know both procedures, the critical value approach and the p-value approach; they must give the same conclusion, and understanding both helps you interpret either kind of output — software prints p-values, but the table-and-critical-value route shows where they come from.
In a research write-up, prefer the phrase "do not reject the null" over "accept the null"; in the exam both phrasings mean the same conclusion.

Key Industry Applications

Every real-world connection named during the session, gathered in one place.

Real-world: microwave radiation safety standards — the 10 microwatts per square centimeter exposure limit used to monitor pollution from large television transmitters. Environmental regulators decide whether a transmitter may keep operating by testing the claim that average exposure stays within the limit.
Real-world: steering system design — low-effort steering for vans modified for disabled drivers, tested by whether the required turning force drops below 54 ounces. Ergonomic design validation for accessibility.
Real-world: computer system performance — measuring whether average turnaround time changes when terminals and printers are added, a before-and-after comparison for IT capacity planning.
Real-world: building materials quality control — verifying the claimed thermal conductivity of cement bricks from 35 determinations; a batch that fails the two-sided test at the claimed 0.340 would be pulled for investigation.
Real-world: wine production — glycerol, a by-product of ethanol fermentation, contributes sweetness, body, and fullness; production control checks the true average concentration against the desired value of 4 with a one-sample t test on a handful of samples per lot.
Real-world: statistical software — Minitab prints the mean, standard deviation, t value, and p-value in one output; R is the free software used to show outputs. The skill of extracting the p-value from printed output is the core exam question pattern.
Real-world: pharmaceutical testing — comparing old and new drugs on heartbeat response, where the variance, not the average, revealed the difference: both drugs averaged 70 beats per minute, but the new drug's extreme reactions were exposed by the spread, not the mean. The same variance logic extends to manufacturing tolerances, where process consistency matters as much as the average.

ASM Lecture 4 notes · Hypothesis Testing Procedures

Advanced Statistical Methods· postgraduate· 2026-08-11

Sections Breakdown

14.1 Why Hypothesis Testing Exists

Two conflicting statements (the null H0 and the alternative HA), the two types of error (type one, alpha; type two, beta), and why decisions from a sample need a controlled procedure.

24.2 Writing the Null and Alternative Hypotheses

The core guidelines for building H0 and HA, with three worked situations: microwave radiation limits, power steering for vans, and computer system turnaround time.

34.3 One-Tailed and Two-Tailed Tests

How the inequality in the alternative fixes the test type - right-tailed, left-tailed, or two-sided - and where the rejection and acceptance regions sit on the curve.

44.4 The Z Test for a Population Mean

The z statistic for a population mean when sigma is known, its sampling-distribution justification, and the five-step hypothesis testing procedure.

54.5 Worked Example: Thermal Conductivity of Cement Bricks

A complete run of the five-step z test at alpha = 0.05: hypotheses, z = 1.77, critical limits +/-1.96, and the verdict to fail to reject the null.

64.6 The P-Value Approach

What a p-value is, computing p-values for z = 1.77, 2.10, and -1.75, the p <= alpha decision rule, and the evidence-strength ladder for interpreting p-values.

74.7 The One-Sample T Test

Choosing between z and t, the t statistic with n - 1 degrees of freedom, the glycerol-in-wine worked example, and reading the Minitab output.

84.8 The Road Ahead: Variance and the Chi-Square Distribution

Why variance matters when averages hide differences, the chi-square statistic (n-1)S^2/sigma^2, and what the next session covers.

9Exam Guidance Summary

Quiz and exam strategy from the session: the five percent default alpha, the p-value rule, output-reading question patterns, and the chi-square applications expected in the mid-semester exam.

10Key Industry Applications

The real-world connections named during the session: safety-limit monitoring, materials quality control, wine production, statistical software output, and pharmaceutical testing.

Postgraduate students in Advanced Statistical Methods

Exam Revision Notes

Below is the distilled, exam-ready core. Every entry comes from the full explanation above. Use this section for rapid review; return to the main notes when a point needs more context.

Why Hypothesis Testing Exists

Must-know: Every test names two statements: H0 (default, equality lives here) and HA (claim to be detected). A decision from data has two possible errors: type one = reject H0 when true (alpha, level of significance), type two = accept H0 when false (beta).

⚠️ Top pitfall: Swapping the two errors: type one is rejecting a true H0 (controlled by alpha); type two is accepting a false H0 (beta). 'Fail to reject H0' does not prove H0 true.

Self-check: If we set alpha = 0.05 and then reject H0, which error are we risking, and what is its probability?

Connects to: 4.2, 4.3.

Writing the Null and Alternative Hypotheses

Must-know: Whatever is to be detected or supported goes in the alternative hypothesis; the null is its negation and the equality sign always lives in the null. One-sided pairs may write the null with the plain equality (e.g. H0: mu = 10) or the inequality (mu <= 10); two-sided tests use H0: mu = mu0 against HA: mu != mu0.

$H_{A} : μ > 10, H_{0} : μ \leq 10$

⚠️ Top pitfall: Putting the equality sign in the alternative, or writing the direction backwards (claiming a reduction but testing >). The two statements must be exact logical opposites.

Self-check: A manufacturer wants to show its new process produces less waste than the old 20 kg per day average. Write H0 and HA.

Connects to: 4.1, 4.3.

One-Tailed and Two-Tailed Tests

Must-know: The alternative decides the test: H_A: mu > mu0 is right-tailed, H_A: mu < mu0 is left-tailed, H_A: mu != mu0 is two-sided. Rejection region is the tail(s) with total area alpha; one-sided critical value z_alpha, two-sided +/- z_alpha/2 (e.g. +/-1.96 at alpha=0.05).

$z \geq z_{α}$

⚠️ Top pitfall: Using a one-sided critical value in a two-sided test, or confusing z_alpha (tail area alpha) with z_alpha/2 (tail area alpha/2).

Self-check: A test has H_A: mu > 10. Which tail holds the rejection region, and what is its area?

Connects to: 4.2, 4.4.

The Z Test for a Population Mean

Must-know: z = (x-bar - mu)/(sigma/root n) with x-bar the sample mean, mu the claimed population mean under H0, sigma the known population standard deviation, n the sample size. The five-step procedure runs hypotheses, alpha (default 0.05 if not given), test statistic, critical value or p-value, conclusion.

$z = \frac{x ˉ - μ}{σ / n}$

⚠️ Top pitfall: Plugging the sample standard deviation s into a z test (that is the t test), putting root n in the numerator, or using the sample mean as the hypothesized mu.

Self-check: A sample of 36 has mean 52; the claimed mean is 50 and sigma = 6. What is the z statistic?

Connects to: 4.3, 4.5, 4.7.

Worked Example: Thermal Conductivity of Cement Bricks

Must-know: The two-sided critical value approach: reject H0 if z >= z_alpha/2 or z <= -z_alpha/2. With alpha = 0.05 the limits are +/-1.96; z = 1.77 lies inside the acceptance region, so we fail to reject H0. The standard error is sigma/root n = 0.00169 here.

$z = \frac{0.343 - 0.340}{0.010/ 35} = \frac{0.003}{0.00169} \approx 1.77$

⚠️ Top pitfall: Rounding the standard error too early, comparing a positive z with the negative critical value, or reading 'fail to reject' as 'proved the claim true'.

Self-check: In the brick test, why is z = 1.77 compared with 1.96 and not with -1.96?

Connects to: 4.3, 4.4, 4.6.

The P-Value Approach

Must-know: p-value = tail area beyond the observed statistic (doubled for a two-sided alternative). Reject H0 when p <= alpha; fail to reject when p > alpha. Both approaches always agree with the critical value approach. Ladder: p < 0.01 overwhelming evidence that HA is true; 0.01-0.05 strong; 0.05-0.10 weak; > 0.10 insufficient.

$p-value = 2 \times P (Z \geq 1.77) = 2 \times 0.0384 = 0.0768$

⚠️ Top pitfall: Getting the direction backwards (p > alpha means fail to reject, not reject), or doubling a p-value for a one-sided test. The p-value is not the probability that H0 is true.

Self-check: A two-sided test gives z = -1.75. What is the p-value and the verdict at alpha = 0.05?

Connects to: 4.5, 4.4, 4.7.

The One-Sample T Test

Must-know: Choice rule: n > 30 with sigma known uses z; small n with sigma unknown uses t = (x-bar - mu)/(s/root n) with n - 1 degrees of freedom. Glycerol test: df 4, critical +/-2.776, t = -0.58 in the acceptance region, p = 0.594 > 0.05 and > 0.10, accept H0. Output reading: extract the p-value (last number) and compare with alpha.

$t = \frac{x ˉ - μ}{s / n}, s = \frac{\sum _{i = 1}^{n} ( x _{i} - x ˉ ) ^{2}}{n - 1}$

⚠️ Top pitfall: Using s in a z test (or sigma in a t test), dividing by n instead of n - 1 in the sample standard deviation, or forgetting the square root. The p-value approach is identical for t and z.

Self-check: Why does the t statistic for the glycerol test have 4 degrees of freedom, and what are the two critical values at alpha = 0.05?

Connects to: 4.4, 4.6, 4.8.

The Road Ahead: Variance and the Chi-Square Distribution

Must-know: Variance hypotheses use sigma^2 != / > / < sigma_0^2 with the chi-square statistic chi^2 = (n-1)S^2/sigma^2, chi-square distributed with n-1 degrees of freedom for a normal population. Unlike z and t it is asymmetric and never negative. Any one of the three chi-square applications (goodness of fit, independence, equality of proportions) can appear in the mid-semester exam.

$χ^{2} = \frac{( n - 1 ) S ^{2}}{σ ^{2}}$

⚠️ Top pitfall: Expecting a symmetric distribution or negative values for chi-square; the statistic is a ratio of squared quantities, always >= 0, with the rejection region in the right tail.

Self-check: Why can the chi-square statistic never be negative, and what is its distribution and degrees of freedom for a sample of size n from a normal population?

Connects to: 4.2, 4.7.

Exam Guidance Summary

Must-know: If alpha is not given, use 0.05. Reject when p <= alpha, fail to reject when p > alpha. Quiz one: 16-30 August, single attempt, chapters 6-8, 25 objective questions at five marks each. Most test questions hand you output and ask you to extract the p-value and conclude.

⚠️ Top pitfall: Reversing the p-value rule (treating a large p-value as grounds to reject) and forgetting the five percent default when alpha is not stated.

Self-check: An output shows p-value = 0.028 with alpha = 0.05. What is the conclusion?

Connects to: 4.4, 4.6, 4.7.

Key Industry Applications

Must-know: Every hypothesis test in this lecture maps to an industry use: safety-limit monitoring (microwave), product design validation (steering), IT performance checks (turnaround time), materials QC (cement bricks), food production control (glycerol in wine), pharmaceutical testing (variance separating two drugs).

⚠️ Top pitfall: Reporting a statistical verdict without its context — a conclusion is only meaningful when tied back to the real-world claim being tested.

Self-check: Which example shows that averages can hide differences that variance exposes?

Connects to: 4.2, 4.5, 4.7, 4.8.

Previous LectureLecture 03 · Statistical Inference and Hypothesis Testing

Next LectureLecture 05 · Inferences on the Population Variance

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Prerequisite Knowledge

Previously Covered in This Subject

4.1 Why Hypothesis Testing Exists

4.1.1 Two Conflicting Statements

4.1.2 The Two Types of Error

4.2 Writing the Null and Alternative Hypotheses

4.2.1 The Core Guidelines

4.2.2 Situation 1: Microwave Radiation

4.2.3 Situation 2: Power Steering for Vans

4.2.4 Situation 3: Computer System Turnaround Time

4.2.5 The Three Situations Side by Side

4.3 One-Tailed and Two-Tailed Tests

4.3.1 The Three Test Types

4.3.2 Rejection and Acceptance Regions

4.4 The Z Test for a Population Mean

4.4.1 The Test Statistic and Its Justification

4.4.2 The Five-Step Procedure

4.5 Worked Example: Thermal Conductivity of Cement Bricks

4.5.1 Problem Setup and Hypotheses

4.5.2 The Critical Value Approach

4.5.3 Student Questions and Answers

4.5.4 The Verdict and Its Meaning

4.6 The P-Value Approach

4.6.1 What the P-Value Is

4.6.2 Computing P-Values: The Cement Brick Case

4.6.3 Worked Cases: z = 2.10 and z = -1.75

4.6.4 Guidelines for Interpreting P-Values

4.6.5 Student Questions and Answers

4.7 The One-Sample T Test

4.7.1 Choosing Between Z and T

4.7.2 The Sample Mean and Sample Standard Deviation

4.7.3 Worked Example: Glycerol in Wine

4.7.4 Reading the Minitab Output

4.7.5 Student Questions and Answers

4.8 The Road Ahead: Variance and the Chi-Square Distribution

4.8.1 Why Variance Matters

4.8.2 The Chi-Square Statistic

4.8.3 What Comes Next

Exam Guidance Summary

Key Industry Applications

ASM Lecture 4 notes · Hypothesis Testing Procedures

Sections Breakdown

Exam Revision Notes

Why Hypothesis Testing Exists

Writing the Null and Alternative Hypotheses

One-Tailed and Two-Tailed Tests

The Z Test for a Population Mean

Worked Example: Thermal Conductivity of Cement Bricks

The P-Value Approach

The One-Sample T Test

The Road Ahead: Variance and the Chi-Square Distribution

Exam Guidance Summary

Key Industry Applications

Was this lecture useful?

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