46 Hypothesis Testing

46.1 What is a Hypothesis Test?

A hypothesis test is a formal statistical procedure for using sample data to decide between two competing claims about a population parameter (gupta2021?; kothari2019?). The framework was put on its modern footing by Jerzy Neyman and Egon Pearson (1933) and Ronald A. Fisher.

46.2 The Two Hypotheses

Null and Alternative Hypotheses

Hypothesis	Symbol	Stand
Null hypothesis	$H_0$	The status quo; no effect, no difference
Alternative hypothesis	$H_1$ or $H_a$	What the researcher hopes to establish

The hypotheses must be mutually exclusive and exhaustive. The test seeks evidence to reject $H_0$ in favour of $H_1$; absence of evidence is not proof of $H_0$.

46.3 Type I and Type II Errors

A test never gives certainty; two kinds of error are possible:

Type I and Type II Errors

Decision \ Truth	$H_0$ true	$H_0$ false
Reject $H_0$	Type I error ($\alpha$)	Correct ($1 - \beta$, power)
Do not reject $H_0$	Correct ($1 - \alpha$)	Type II error ($\beta$)

$\alpha$ — Significance level; probability of rejecting a true $H_0$. Conventional values: 0.05, 0.01, 0.10.
$\beta$ — Probability of failing to reject a false $H_0$.
Power = $1 - \beta$ — probability of correctly rejecting a false $H_0$. Larger sample size raises power.

46.4 One-Tailed vs Two-Tailed Tests

One-Tailed vs Two-Tailed Tests

Test	Form of $H_1$	Critical region
Two-tailed	$\mu \neq \mu_0$	Both tails (split $\alpha$ = $\alpha/2$ each side)
Right-tailed	$\mu > \mu_0$	Right tail only
Left-tailed	$\mu < \mu_0$	Left tail only

46.5 Procedure of Hypothesis Testing

Six Steps in Hypothesis Testing

Step	Action
1	Formulate $H_0$ and $H_1$
2	Choose the level of significance $\alpha$
3	Identify the appropriate test statistic and its sampling distribution
4	Compute the test statistic from the sample
5	Compare with the critical value (or compute p-value)
6	Take a decision and state the conclusion

46.6 Critical Value vs p-Value Approach

Two equivalent decision rules:

Critical-value approach — reject $H_0$ if the test statistic falls in the critical region.
p-value approach — reject $H_0$ if $p < \alpha$.

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one observed, assuming $H_0$ is true.

46.7 Major Tests — A Compact Map

Common Hypothesis Tests

Situation	Test	Statistic
Test of mean, $\sigma$ known, large $n$	Z-test	$z = \dfrac{\bar X - \mu_0}{\sigma/\sqrt n}$
Test of mean, $\sigma$ unknown, small $n$	t-test	$t = \dfrac{\bar X - \mu_0}{s/\sqrt n}$
Test of two means	Independent / paired t-test	$t = \dfrac{\bar X_1 - \bar X_2}{\text{SE}}$
Test of proportion	Z-test	$z = \dfrac{p - P_0}{\sqrt{P_0(1-P_0)/n}}$
Test of variance / Goodness of fit	$\chi^2$ test	$\chi^2 = \sum \dfrac{(O - E)^2}{E}$
Test of variances or ANOVA	F-test	$F = \dfrac{s_1^2}{s_2^2}$ or MSB / MSW
Independence in contingency table	$\chi^2$ test	$\chi^2 = \sum \dfrac{(O - E)^2}{E}$, df = $(r-1)(c-1)$

46.8 Parametric vs Non-Parametric Tests

Parametric vs Non-Parametric Tests

Family	Assumes population distribution?	Examples
Parametric	Yes — usually normal	t, Z, F, ANOVA, Pearson correlation
Non-parametric	No — distribution-free	Chi-square, Sign test, Wilcoxon, Mann-Whitney, Kruskal-Wallis, Spearman rank, Run test

46.9 ANOVA — Analysis of Variance

R.A. Fisher’s ANOVA tests whether the means of three or more groups are equal. One-way ANOVA compares groups by a single factor; two-way ANOVA by two factors.

The F-statistic:

\[ F = \dfrac{\text{Mean Square Between (MSB)}}{\text{Mean Square Within (MSW)}} \]

A large $F$ indicates that variation between groups dominates variation within groups, leading to rejection of equality.

46.10 Worked Example — Z-test for Mean

A factory claims its bulbs last 1,000 hours on average ($\sigma = 50$). A sample of $n = 100$ bulbs averages 990 hours. Test at $\alpha = 0.05$.

$H_0: \mu = 1000$; $H_1: \mu \neq 1000$ (two-tailed).
$z = (990 - 1000) / (50 / \sqrt{100}) = -10 / 5 = -2$.
Critical value at $\alpha = 0.05$ (two-tailed) = $\pm 1.96$.
$|z| = 2 > 1.96$ → reject $H_0$ — the claim is not supported.

46.11 Exam-Pattern MCQs

Q 01

Rejecting a true null hypothesis is a:

AType I error
BType II error
CSampling error
DTest power

View solution

Correct Option: A

Rejecting a true $H_0$ is a Type I error; its probability is $\alpha$.

Q 02

Match each test with the situation in which it is most appropriate:

	Test		Situation
(i)	Z-test	(a)	Goodness-of-fit
(ii)	t-test	(b)	Mean, $\sigma$ unknown, small $n$
(iii)	$\chi^2$ test	(c)	Mean, $\sigma$ known, large $n$
(iv)	F-test	(d)	Compare three or more group means

A(i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
B(i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
C(i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
D(i)-(d), (ii)-(a), (iii)-(b), (iv)-(c)

View solution

Correct Option: A

Q 03

The power of a test equals:

A$\alpha$
B$\beta$
C$1 - \alpha$
D$1 - \beta$

View solution

Correct Option: D

Power = probability of correctly rejecting a false $H_0$ = $1 - \beta$.

Q 04

A two-tailed Z-test at 5 % level rejects $H_0$ if:

A$|z| > 1.645$
B$|z| > 1.96$
C$|z| > 2.33$
D$|z| > 2.58$

View solution

Correct Option: B

The two-tailed 5 % critical Z is ±1.96.

Q 05

Match the test family with the assumption it makes:

	Family		Assumption
(i)	Parametric	(a)	No specific distributional assumption
(ii)	Non-parametric	(b)	Population distribution (usually normal)

A(i)-(b), (ii)-(a)
B(i)-(a), (ii)-(b)

View solution

Correct Option: A

Q 06

A sample of 100 bulbs averages 990 hours; population claim is 1,000 hours with $\sigma = 50$. The Z value is:

A−1.0
B−2.0
C−5.0
D−10.0

View solution

Correct Option: B

$z = (990 − 1000)/(50/\sqrt{100}) = −10/5 = $ −2.

Q 07

Arrange the steps of hypothesis testing in correct order: (i) Choose level of significance (ii) Formulate $H_0$ and $H_1$ (iii) Compare with critical value (iv) Compute test statistic

A(ii), (i), (iv), (iii)
B(i), (ii), (iii), (iv)
C(iv), (iii), (ii), (i)
D(iii), (iv), (ii), (i)

View solution

Correct Option: A

Q 08

Match each non-parametric test with its parametric analogue:

	Non-parametric		Analogue
(i)	Wilcoxon signed-rank	(a)	Pearson correlation
(ii)	Mann-Whitney	(b)	Paired t-test
(iii)	Kruskal-Wallis	(c)	Independent t-test
(iv)	Spearman rank	(d)	One-way ANOVA

A(i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
B(i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
C(i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
D(i)-(d), (ii)-(a), (iii)-(c), (iv)-(b)

View solution

Correct Option: A

Quick recall

Hypothesis test — compares $H_0$ (status quo) and $H_1$ (researcher’s claim) using sample data.
Type I error (reject true $H_0$, $\alpha$); Type II error (fail to reject false $H_0$, $\beta$). Power = 1 − $\beta$.
6 steps: Formulate → Choose $\alpha$ → Identify test → Compute → Compare → Decide.
Tests: Z (mean, $\sigma$ known, large $n$); t (mean, $\sigma$ unknown, small $n$); $\chi^2$ (goodness-of-fit, independence); F / ANOVA (variances, three+ means); Z for proportion.
95 % two-tailed critical Z = ±1.96. 99 % = ±2.58. 90 % = ±1.645.
$\chi^2$: $\sum (O - E)^2 / E$, df = $(r-1)(c-1)$ for contingency.
ANOVA F = MSB / MSW.
Parametric assumes distribution (usually normal); non-parametric does not.
Non-parametric ↔︎ parametric: Wilcoxon ↔︎ paired t; Mann-Whitney ↔︎ independent t; Kruskal-Wallis ↔︎ one-way ANOVA; Spearman ↔︎ Pearson.
p-value = probability of test statistic at least as extreme, given $H_0$. Reject if p < $\alpha$.

Decision \ Truth	\(H_0\) true	\(H_0\) false
Reject \(H_0\)	Type I error (\(\alpha\))	Correct (\(1 - \beta\), power)
Do not reject \(H_0\)	Correct (\(1 - \alpha\))	Type II error (\(\beta\))