Statistics 2nd Year Chapter 13 Hypothesis Testing Notes (ICS Part-II)

Chapter 12 answered the question 'what is a plausible range of values for an unknown parameter?' through confidence intervals. This chapter answers a different but closely related question: 'is a specific claimed value of that parameter believable, given what the sample shows?' This is hypothesis testing — the second and last major branch of statistical inference, and by far the most heavily used tool in applied statistics, business decision-making, and scientific research. It is also the largest chapter in the book (70 pages), because it methodically works through the same battery of situations covered in Chapter 12 — mean, proportion, difference of two means (independent and paired/dependent samples), difference of two proportions — but now framed as formal decision procedures rather than interval estimates.

The chapter builds a single, disciplined framework — state hypotheses, choose a significance level, pick a test statistic, define a rejection region, compute the observed value, and draw a conclusion — and then applies that exact same framework, unchanged, to every situation that follows. Master the framework once in section 13.1, and the rest of the chapter is just plugging different test statistics (Z or T) into the same seven-step procedure.

Learning Objectives

• Define statistical hypothesis, null hypothesis, and alternative hypothesis, and correctly formulate them from a verbal claim
• Distinguish simple from composite hypotheses, and one-tailed from two-tailed tests
• Define Type-I and Type-II errors, and the level of significance and level of confidence
• List and apply the seven standard steps of a hypothesis test
• Test hypotheses about a population mean using the Z and T test statistics, choosing correctly between them
• Test hypotheses about a population proportion, with and without the continuity correction
• Test hypotheses about the difference between two population means for independent samples (known variance, pooled t, large-sample Z)
• Test hypotheses about the difference between two population means for dependent (paired/matched) samples
• Test hypotheses about the difference between two population proportions

Key Concepts

13.1 The Elements of a Test of Hypothesis

A statistical hypothesis is an assertion or conjecture about the distribution of one or more random variables — typically phrased quantitatively in terms of a population parameter (e.g., mu=25, pi=0.4). Hypothesis testing is the procedure used to determine whether such an assumption is supported by an observed random sample. The null hypothesis, H0, is the hypothesis tested for possible rejection under the assumption that it is true — it is always a statement of 'no effect' or 'the status quo' (a coin is unbiased, a drug is ineffective, there is no difference between two methods). The alternative hypothesis, H1, is the hypothesis we are willing to accept if H0 is rejected — it is often called the research hypothesis, because it expresses what the researcher actually believes and is trying to support with evidence.

Formulating H0 and H1 correctly follows a fixed procedure: first state the alternative hypothesis as whatever the researcher is trying to find support for (an inequality: 'less than', 'greater than', or 'not equal to'), then state the null hypothesis as the complementary claim with an equality sign built in. The three matched pairs are: H0: theta>=theta0 vs H1: theta<theta0 (left-tailed); H0: theta<=theta0 vs H1: theta>theta0 (right-tailed); H0: theta=theta0 vs H1: theta!=theta0 (two-tailed). A simple hypothesis completely specifies the population distribution (both its functional form and every parameter value); a composite hypothesis does not (e.g., specifying only the mean while leaving the variance unspecified or as an inequality).

The test statistic is the sample statistic used to decide whether to reject H0 — commonly Z, T, chi-square, or F. The rejection (critical) region is the set of test-statistic values for which H0 is rejected; the nonrejection region is the complementary set; the critical values are the boundary points separating them. A two-tailed test places the rejection region equally in both tails of the sampling distribution (used when H1 is theta!=theta0); a one-tailed test places it entirely in one tail — a left-tailed test when H1 is theta<theta0 (looking for a definite decrease), and a right-tailed test when H1 is theta>theta0 (looking for a definite increase).

Because a decision is being made from incomplete (sample) information, two kinds of error are possible. A Type-I error is rejecting H0 when H0 is actually true. A Type-II error is accepting (not rejecting) H0 when H1 is actually true. The probability of a Type-I error is denoted alpha and is called the level of significance (also the size of the critical region/test) — a small, pre-assigned value such as 0.05 or 0.01, fixed BEFORE the sample is drawn so the results cannot bias the choice. The probability of a Type-II error is denoted beta. The level of confidence, 1-alpha, is the probability of correctly accepting a true H0. A hypothesis test concludes either by rejecting H0 in favour of H1 (if the test statistic falls in the rejection region), or by failing to reject H0 (there is insufficient evidence to reject it — this is NOT the same as proving H0 true).

13.1.25 The Seven Steps of a Hypothesis Test

Every hypothesis test in this chapter follows the identical disciplined sequence. Before looking at the sample: (1) identify the population and state the conditions required for the test's validity; (2) formulate and state H0 and H1; (3) decide and specify the level of significance, alpha; (4) select the appropriate test statistic and its sampling distribution assuming H0 is true; (5) find the critical value(s) of the test statistic for the chosen alpha; (6) establish the rejection region; (7) state the decision rule (reject H0 if the test statistic falls in the rejection region, otherwise do not reject). After the sample is collected: (8) calculate the observed value of the test statistic from the sample; (9) draw the conclusion — reject or do not reject H0 — and state it in plain, practical terms.

13.2 Test of Hypothesis About a Population Mean, mu

The three possible hypothesis pairs about mu (against a hypothesised value mu0) are H0: mu>=mu0 vs H1: mu<mu0; H0: mu<=mu0 vs H1: mu>mu0; and H0: mu=mu0 vs H1: mu!=mu0. The choice of test statistic mirrors Chapter 12's confidence-interval logic exactly: when the population is normal with KNOWN variance sigma^2 (any sample size), use Z = (X-bar – mu0)/(sigma/sqrt(n)), which is exactly standard normal under H0. When the population is normal with UNKNOWN variance and the sample is SMALL (n<=30), use T = (X-bar – mu0)/(S-hat/sqrt(n)), which follows a t-distribution with v=n-1 degrees of freedom under H0. When the sample is LARGE (n>30), the Central Limit Theorem justifies using Z = (X-bar – mu0)/(S-hat/sqrt(n)) as an approximation regardless of the population's shape or whether sigma is known.

In every case the mechanics are the same: compute the observed value of the test statistic from the sample, compare it against the critical value(s) that mark off the rejection region at the chosen alpha, and reject H0 if the observed value falls in that region.

13.3 Test of Hypothesis About a Population Proportion, pi

For a hypothesised proportion pi0, the three hypothesis pairs are analogous to those for the mean. Since the sampling distribution of the sample proportion P=X/n is (for large n) approximately normal with mean pi0 and variance pi0(1-pi0)/n, the test statistic is Z = (P – pi0)/sqrt[pi0(1-pi0)/n], or equivalently in terms of the count X: Z = (X – n.pi0)/sqrt[n.pi0(1-pi0)]. For extra precision, a continuity correction can be applied (since X is discrete but is being approximated by a continuous normal distribution): Z = [(X +/- 0.5) – n.pi0]/sqrt[n.pi0(1-pi0)] — using a plus sign when X<n.pi0 (or P<pi0) and a minus sign when X>n.pi0 (or P>pi0).

13.4 Difference Between Two Means — Independent Samples

For comparing mu1 and mu2 via independent samples, against a hypothesised difference delta0, three cases parallel Chapter 12 exactly. Known variances (any sample size, normal populations): Z = [(X-bar-1 – X-bar-2) – delta0] / sqrt[sigma1^2/n1 + sigma2^2/n2], exactly standard normal under H0. Small samples (n1,n2<=30), normal populations, equal but unknown variance: first pool the two sample variances into Sp^2 = [(n1-1).S-hat-1^2 + (n2-1).S-hat-2^2]/(n1+n2-2), then use T = [(X-bar-1 – X-bar-2) – delta0] / [Sp.sqrt(1/n1 + 1/n2)], which follows a t-distribution with v=n1+n2-2 degrees of freedom. Large samples (n1,n2>30), any population shape: the CLT justifies Z = [(X-bar-1 – X-bar-2) – delta0] / sqrt[S-hat-1^2/n1 + S-hat-2^2/n2] as an approximation, with no assumption of equal variances required.

13.5 Difference Between Two Means — Dependent (Paired) Samples

When observations naturally come in matched pairs (before/after measurements on the same subject, or two products tested on the same set of items), the two samples are dependent rather than independent — pairing lets each subject act as its own control, filtering out subject-to-subject variability that would otherwise inflate the noise in an independent-samples comparison. For each pair, define the difference D = Y – X (after minus before, or Candidate-A minus Candidate-B, etc.). The population of these differences has mean mu_D = mu2 – mu1 and its own variance sigma_D^2 — and testing mu_D reduces the two-sample problem to an ordinary ONE-sample test on the differences, using exactly the machinery from section 13.2.

Given n paired differences d1, d2, …, dn with mean d-bar = (sum d_i)/n and standard deviation s-hat_D = sqrt{[sum(d_i – d-bar)^2]/(n-1)}, the test statistic for H0: mu_D = delta0 (delta0 is usually 0, i.e. 'no difference') is T = (D-bar – delta0)/(s-hat_D/sqrt(n)), following a t-distribution with v=n-1 degrees of freedom. Two assumptions are required: the n differences must be a random sample from the population of differences, and that population of differences must be (approximately) normally distributed — though this normality assumption can be relaxed for large n (>30) by the Central Limit Theorem, exactly as with a single-sample mean test.

13.6 Difference Between Two Proportions

For comparing pi1 and pi2 from two independent samples (sizes n1, n2, sample proportions P1=X1/n1, P2=X2/n2), two distinct situations arise. To test the GENERAL hypothesis H0: pi1-pi2=delta0 for some non-zero delta0, use Z = [(P1-P2) – delta0] / sqrt[P1(1-P1)/n1 + P2(1-P2)/n2], using the two sample proportions separately in the standard error. To test specifically whether the two populations have the SAME proportion of successes, H0: pi1=pi2 (i.e. delta0=0), it is more efficient to POOL the two samples into a single combined estimate of the common proportion, pi-hat = (X1+X2)/(n1+n2) = (n1.P1+n2.P2)/(n1+n2), and use Z = (P1-P2) / sqrt{pi-hat(1-pi-hat).(1/n1 + 1/n2)} — exactly mirroring the pooled-variance logic used for two means with equal variance in section 13.4.

Important Definitions

Key Facts and Relations

Topic	Key Fact / Relation
Test statistic for mu, sigma known / large n	Z = (X-bar – mu0) / (sigma/sqrt(n)) [or S-hat/sqrt(n) if sigma unknown, large n]
Test statistic for mu, small n, normal, sigma unknown	T = (X-bar – mu0) / (S-hat/sqrt(n)), v = n-1
Test statistic for proportion pi	Z = (P – pi0) / sqrt[pi0(1-pi0)/n]
Continuity-corrected Z for proportion	Z = [(X +/- 0.5) – n.pi0] / sqrt[n.pi0(1-pi0)]
Difference of means, known variances	Z = [(Xbar1-Xbar2) – delta0] / sqrt[sigma1^2/n1 + sigma2^2/n2]
Difference of means, small n, pooled variance	T = [(Xbar1-Xbar2) – delta0] / [Sp.sqrt(1/n1+1/n2)], v=n1+n2-2
Pooled variance, Sp^2	Sp^2 = [(n1-1)S1^2 + (n2-1)S2^2] / (n1+n2-2)
Difference of means, large samples	Z = [(Xbar1-Xbar2) – delta0] / sqrt[S1^2/n1 + S2^2/n2]
Paired-sample test (mu_D)	T = (D-bar – delta0) / (S_D-hat/sqrt(n)), v = n-1; D-bar = (sum d_i)/n
Difference of proportions, general delta0	Z = [(P1-P2) – delta0] / sqrt[P1(1-P1)/n1 + P2(1-P2)/n2]
Difference of proportions, testing pi1=pi2 (pooled)	Z = (P1-P2) / sqrt{pi-hat(1-pi-hat)(1/n1+1/n2)}, pi-hat=(X1+X2)/(n1+n2)
Error probabilities	alpha = P(reject H0 | H0 true) = P(Type-I); beta = P(accept H0 | H1 true) = P(Type-II)

Diagrams

One-Tailed vs Two-Tailed Rejection Regions: Three standard normal curves side by side showing the shaded rejection region for a left-tailed test (Z<-z_alpha), a right-tailed test (Z>z_alpha), and a two-tailed test (split equally in both tails, Z<-z_alpha/2 or Z>z_alpha/2), illustrating how the shape of H1 determines where the critical region is placed

Type-I and Type-II Error Trade-off: A two-panel diagram showing the sampling distribution of the test statistic under H0 and under H1 side by side (overlapping), with the alpha region (Type-I error, area under the H0 curve past the critical value) and the beta region (Type-II error, area under the H1 curve on the non-rejection side of the critical value) shaded and labelled, illustrating the fundamental trade-off between the two error types

Worked Z-Test Example: Coffee Jar Weight: A standard normal curve for Example 13.6 (H0: mu>=6.0oz vs H1: mu<6.0oz, alpha=0.05) with the left-tail rejection region (Z<-1.645) shaded and the observed test statistic z=-5.0 plotted well inside the rejection region, visually confirming the decision to reject H0

Short Questions & Answers

Long Questions & Answers

Explain the complete logical structure of a hypothesis test — the null and alternative hypotheses, Type-I and Type-II errors, the level of significance, and the seven-step testing procedure — using a concrete example to illustrate each concept.

What are the null and alternative hypotheses, and how must they be formulated?

The null hypothesis, H0, is a statement of 'no effect' or the status quo — the claim put on trial and assumed true unless the evidence strongly contradicts it. The alternative hypothesis, H1, is the competing claim the researcher actually suspects, often called the research hypothesis. The two must be formulated as an exhaustive, mutually exclusive pair: by convention H0 carries the equality sign (>=, <=, or =) while H1 takes the strict complement (<, >, or !=).

What are Type-I and Type-II errors, and what is the level of significance?

A Type-I error occurs when H0 is rejected even though it was actually true — a false alarm. A Type-II error occurs when H0 is NOT rejected even though H1 was actually true — a missed detection. The level of significance, alpha, is the small, pre-chosen probability of a Type-I error, fixed before any data is collected so it cannot be influenced by the results. Its complement, 1 – alpha, is the level of confidence, the probability of correctly failing to reject a true H0.

What is a test statistic, and how does the shape of H1 determine a one-tailed versus two-tailed test?

A test statistic is a sample-based quantity, such as Z or T, whose sampling distribution under the assumption that H0 is true is known and can be looked up. When H1 specifies a direction (using < or >), the rejection region is placed entirely in one tail of that distribution — a one-tailed test. When H1 simply states 'not equal to', with no direction specified, the rejection region is split equally between both tails — a two-tailed test.

Walk through the coffee-jar example: how are H0, H1, and the rejection region set up?

A machine claims to fill jars with a mean of 6.0 ounces, and the concern is underfilling, so H1: mu < 6.0 (a left-tailed test) with H0: mu >= 6.0. Alpha is set at 0.05 before a random sample of 100 jars is weighed. Since the population standard deviation is known, the test statistic is Z = (X-bar – mu0)/(sigma/sqrt(n)), standard normal under H0. The left-tailed critical value at alpha=0.05 is z=-1.645, so the rejection region is Z < -1.645.

What is the conclusion of the coffee-jar test, and what would each type of error mean in that context?

With an observed sample mean of 5.9 ounces, the test statistic is z = (5.9-6.0)/(0.2/sqrt(100)) = -5.0, far beyond the critical value of -1.645, so it falls in the rejection region and H0 is rejected: the machine is underfilling at the 5% significance level. A Type-I error here would mean concluding the machine underfills when it actually doesn't; a Type-II error would mean failing to detect real underfilling, leaving customers shortchanged.

Compare and contrast the test of the difference between two means for independent samples versus dependent (paired) samples, explaining when each design is appropriate, how each test statistic is constructed, and why pairing can be advantageous.

What is the difference between independent and dependent (paired) samples?

Two samples are independent if selecting observations from one population has no bearing on selecting observations from the other — for example, separate samples drawn from two different machines. Two samples are dependent, or paired/matched, if each observation in one sample is deliberately linked to a specific corresponding observation in the other, forming genuine pairs — the classic example being the same individuals measured before and after a treatment, such as weight before and after a diet programme.

What are the three test-statistic cases for comparing two independent means?

When both populations are normal with known variances, Z = [(X-bar1 – X-bar2) – delta0] / sqrt(sigma1^2/n1 + sigma2^2/n2) is exactly standard normal for any sample size. When both samples are small (n1, n2 <= 30), populations normal, and variances unknown but assumed equal, the variances are pooled as Sp^2 = [(n1-1)S1hat^2 + (n2-1)S2hat^2]/(n1+n2-2), giving T = [(X-bar1-X-bar2)-delta0]/[Sp*sqrt(1/n1+1/n2)], a t-distribution with v=n1+n2-2 df. When both samples are large (>30), the Central Limit Theorem removes the normality and equal-variance requirements, giving Z = [(X-bar1-X-bar2)-delta0]/sqrt(S1hat^2/n1+S2hat^2/n2), using each sample's own variance.

How is the test for paired (dependent) samples constructed?

Each matched pair is collapsed into a single difference, D = Y – X (for example, weight after minus weight before). This turns the two-sample problem into an ordinary one-sample problem about mu_D = mu2 – mu1. The test statistic T = (D-bar – delta0)/(SDhat/sqrt(n)), calculated from the sample mean and standard deviation of the n differences, follows a t-distribution with v=n-1 degrees of freedom — exactly mirroring the single-sample small-sample t-test.

Why does pairing improve statistical power compared to an independent-samples design?

Pairing lets each subject act as their own control, stripping out subject-to-subject variability — differences in starting weight, metabolism, and so on — that would otherwise inflate the noise in an independent-samples comparison. Because each person's baseline characteristics cancel out when the difference D is calculated, only the genuine treatment effect remains, which is why paired designs generally achieve greater statistical power to detect a real difference, especially when sample sizes are small.

Multiple Choice Questions (MCQs)

Quick Revision Summary

• H0 = status quo/no-effect (tested for rejection) | H1 = research claim (accepted only if H0 rejected)
• Type-I error = reject true H0, probability alpha | Type-II error = accept false H0, probability beta
• Left-tailed: H1 uses '<' | Right-tailed: H1 uses '>' | Two-tailed: H1 uses '!='
• 7 steps: state hypotheses -> set alpha -> pick test statistic -> find critical value(s) -> rejection region -> decision rule -> compute & conclude
• Mean test: sigma known -> Z; small n normal sigma unknown -> T (v=n-1); large n -> Z (CLT)
• Proportion test: Z=(P-pi0)/sqrt[pi0(1-pi0)/n]; continuity correction adds/subtracts 0.5
• Diff of means, independent: known var -> Z; small n equal var -> pooled T (v=n1+n2-2); large n -> Z
• Diff of means, paired/dependent: reduces to 1-sample T-test on differences D=Y-X, v=n-1
• Diff of proportions: general delta0 -> separate P1,P2 in SE; testing pi1=pi2 -> pooled pi-hat

Exam Tips

• Always write H0 with the equality sign (>=, <=, or =) and H1 as the strict, direction-matching complement
• Match the tail of the test to H1's inequality: '<' = left tail, '>' = right tail, '!=' = two tails (split alpha/2 each side)
• For mean tests, check THREE things before picking Z or T: is sigma known, is the population normal, is n large or small
• For paired-sample problems, compute the differences FIRST (consistently: always after-minus-before, or always A-minus-B) before doing anything else
• When testing pi1=pi2 specifically, don't forget to pool: use pi-hat in the standard error, not separate p1 and p2
• State your final conclusion in plain managerial language, not just 'reject H0' — explain what that means for the actual question being asked