🧪 Hypothesis Testing

Last Updated: Jan 2026

Hypothesis Testing is a statistical method used to make decisions about a population using sample data.

We test an assumption (hypothesis) and decide whether there is enough evidence to:

• Accept it, or
• Reject it

🗣 Hinglish Tip: Hypothesis testing = ek claim ko data ke through prove ya disprove karna

Used when we want to:

• Compare values
• Check claims
• Validate assumptions
• Make decisions with confidence

Real-World Use Cases

• Medicine → Is new drug effective?
• Manufacturing → Is defect rate acceptable?
• Business → Did sales increase after campaign?
• ML / Data Science → Model performance comparison

Basic Terminology

• Population → Entire data
• Sample → Part of population
• Test Statistic → Value calculated from sample
• Significance Level ($\alpha$) → Probability of rejecting true hypothesis (Common: 0.05)

Symbols Used in Hypothesis Testing:

• $H_0$ = Null Hypothesis/default assumption
• $H_1$ = Alternative Hypothesis
• $\alpha$ = Significance Level
• p-value/critical value = Probability of rejecting true hypothesis

Types of Hypothesis

Every hypothesis test has two hypotheses:

Null Hypothesis ($H_0$)

• Default assumption
• States no effect or no difference

Examples:

$H_0: \mu = 50
H_0: p = 0.4$

🗣 Hinglish Tip: Null hypothesis = kuch bhi change nahi hua

Alternative Hypothesis ($H_1$ or $H_a$)

• Opposite of null hypothesis
• Indicates effect or difference

Examples:

$H_1: \mu \ne 50
\\[0.5em]
H_1: \mu > 50
\\[0.5em]
H_1: \mu < 50$

Types of Alternative Hypothesis (Based on Direction)

Two-Tailed Test

• Tests both sides
• Checks for any difference

$H_0: \mu = \mu_0
\\[0.5em]
H_1: \mu \ne \mu_0$

Use case:

• Quality control
• Equality check

Right-Tailed Test

• Tests greater than
• Focus on right side

$H_0: \mu \le \mu_0
\\[0.5em]
H_1: \mu > \mu_0$

Use case:

• Performance improvement
• Sales increase

Left-Tailed Test

• Tests less than
• Focus on left side

$H_0: \mu \ge \mu_0
\\[0.5em]
H_1: \mu < \mu_0$

Use case:

• Defect reduction
• Cost decrease

Types of Hypothesis Testing (Based on Data & Condition)

Z-Test

Used when:

• Sample size $n \ge 30$
• Population standard deviation ($\sigma$) known
• Data is normal or CLT applies

Use case:

• Large sample testing

t-Test

Used when:

• Sample size $n < 30$
• Population standard deviation unknown

Types:

• One-sample t-test
• Two-sample t-test
• Paired t-test

Use case:

• Small samples

Chi-Square Test ($\chi^2$)

Used for:

• Categorical data
• Independence testing
• Goodness of fit

Use case:

• Gender vs preference
• Survey analysis

ANOVA (F-Test)

Used to:

• Compare more than two means

Use case:

• Multiple group comparison

Hypothesis Testing Workflow

1. State hypotheses ($H_0$ & $H_1$)
2. Choose significance level ($\alpha$)
3. Select appropriate test
4. Compute test statistic
5. Find p-value or critical value
6. Make decision (Reject or Fail to Reject $H_0$)

Decision Rule

• If p-value $\le \alpha$ → Reject $H_0$
• If p-value $> \alpha$ → Fail to reject $H_0$

🗣 Hinglish Tip: Chhota p-value = strong evidence against $H_0$

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