Hypothesis Testing

Introduction to Hypothesis Testing

Statistical Hypothesis: An assumption about a population parameter which may or may not be true.

Null Hypothesis (H₀): A statement of no effect or no difference, which we test for possible rejection.

Alternative Hypothesis (H₁ or Hₐ): A statement that contradicts the null hypothesis, representing what we suspect might be true.

Key Points:

Hypothesis testing is a method to make decisions using data
We always test the null hypothesis
The alternative hypothesis can be one-tailed or two-tailed
Significance level (α) is the probability of rejecting H₀ when it's true

Types of Errors in Hypothesis Testing

Error Type	Definition	Probability	Also Known As
Type I Error	Rejecting H₀ when it's actually true	α (significance level)	False positive
Type II Error	Failing to reject H₀ when it's false	β	False negative

Power of Test (1-β): The probability of correctly rejecting H₀ when it's false.

Important Relationships:

α and β are inversely related for a fixed sample size
Increasing sample size decreases both α and β
Critical region is the set of values that leads to rejecting H₀

Example: In a medical test, H₀: "Patient is healthy". Type I error = Diagnosing a healthy patient as diseased (false positive). Type II error = Failing to detect a diseased patient (false negative).

Z-test for Single Mean

Z-test: Used when we want to test whether the sample mean differs significantly from the population mean, given that the population variance is known and sample size is large (n ≥ 30).

Test Statistic: Z = (x̄ - μ₀) / (σ/√n)
Where:

x̄ = sample mean
μ₀ = hypothesized population mean
σ = population standard deviation
n = sample size

Example: A company claims its light bulbs last 1000 hours. We test 50 bulbs and find x̄ = 980 hours. Population σ = 80 hours. Test at α = 0.05.

H₀: μ = 1000
H₁: μ ≠ 1000 (two-tailed)

Z = (980 - 1000)/(80/√50) = -20/11.31 ≈ -1.77
Critical Z = ±1.96
Since -1.77 > -1.96, we fail to reject H₀.

When to use Z-test:

Population variance is known
Sample size is large (n ≥ 30)
Data is normally distributed or sample size is very large

Z-test for Difference of Means

Z-test for two means: Used to compare two population means when the population variances are known and samples are independent.

Test Statistic: Z = [(x̄₁ - x̄₂) - (μ₁ - μ₂)] / √(σ₁²/n₁ + σ₂²/n₂)
Where:

x̄₁, x̄₂ = sample means
μ₁, μ₂ = population means (often μ₁ - μ₂ = 0 under H₀)
σ₁, σ₂ = population standard deviations
n₁, n₂ = sample sizes

Example: Test if two teaching methods differ in effectiveness. Method A (n=50, x̄=78, σ=8), Method B (n=60, x̄=75, σ=7). α=0.05.

H₀: μ₁ = μ₂
H₁: μ₁ ≠ μ₂

Z = (78-75)/√(8²/50 + 7²/60) = 3/√(1.28 + 0.82) ≈ 3/1.45 ≈ 2.07
Critical Z = ±1.96
Since 2.07 > 1.96, we reject H₀.

Student's t-test for Single Mean

t-test: Used when population variance is unknown and sample size is small (n < 30). Uses sample standard deviation.

Test Statistic: t = (x̄ - μ₀) / (s/√n)
Where:

x̄ = sample mean
μ₀ = hypothesized population mean
s = sample standard deviation
n = sample size
Degrees of freedom = n-1

Example: A car claims 20 km/liter. Test with 10 cars gives x̄=18.5, s=1.5. α=0.05.

H₀: μ = 20
H₁: μ < 20 (one-tailed)

t = (18.5-20)/(1.5/√10) = -1.5/0.474 ≈ -3.16
Critical t (df=9, α=0.05, one-tailed) ≈ -1.833
Since -3.16 < -1.833, we reject H₀.

Key Points:

Use when σ is unknown and n < 30
t-distribution is similar to normal but with heavier tails
As n increases, t approaches z
Always check degrees of freedom

t-test for Difference of Means

Independent Samples t-test

Compares means from two independent groups when variances are unknown and assumed equal.

Test Statistic: t = (x̄₁ - x̄₂) / [sₚ√(1/n₁ + 1/n₂)]
Where pooled standard deviation sₚ = √[((n₁-1)s₁² + (n₂-1)s₂²)/(n₁+n₂-2)]
df = n₁ + n₂ - 2

Paired t-test

Used when samples are dependent (before/after measurements on same subjects).

Test Statistic: t = d̄ / (s_d/√n)
Where:

d̄ = mean of differences
s_d = standard deviation of differences
n = number of pairs
df = n-1

Example (Independent): Compare two teaching methods. Method A (n=15, x̄=75, s=8), Method B (n=12, x̄=70, s=6). α=0.05.

sₚ = √[(14×64 + 11×36)/25] = √[(896+396)/25] ≈ √51.68 ≈ 7.19
t = (75-70)/[7.19√(1/15 + 1/12)] ≈ 5/(7.19×0.39) ≈ 1.79
Critical t (df=25, two-tailed) ≈ ±2.06
Fail to reject H₀.

F-test

F-test: Used to compare two population variances or in ANOVA to compare multiple means.

Test Statistic for Variance Comparison: F = s₁²/s₂² (where s₁² > s₂²)
df₁ = n₁ - 1 (numerator)
df₂ = n₂ - 1 (denominator)

Example: Compare variability of two machines. Machine A (n=10, s=5), Machine B (n=8, s=3). α=0.05.

H₀: σ₁² = σ₂²
H₁: σ₁² ≠ σ₂²

F = 5²/3² = 25/9 ≈ 2.78
Critical F (df₁=9, df₂=7, α=0.05) ≈ 3.68
Since 2.78 < 3.68, fail to reject H₀.

Important Points:

Always put larger variance in numerator
F-distribution is right-skewed
Used to check assumption of equal variances before t-test
Fundamental to ANOVA

Chi-square Test for Goodness of Fit

Chi-square test: Tests whether observed frequencies differ significantly from expected frequencies.

Test Statistic: χ² = Σ[(Oᵢ - Eᵢ)²/Eᵢ]
Where:

Oᵢ = observed frequency
Eᵢ = expected frequency
df = k - 1 - p (k = categories, p = estimated parameters)

Example: Test if die is fair (60 rolls: 8 ones, 12 twos, 9 threes, 11 fours, 10 fives, 10 sixes). α=0.05.

H₀: Die is fair (Eᵢ=10 for all)
H₁: Die is not fair

χ² = (8-10)²/10 + (12-10)²/10 + ... + (10-10)²/10 = 4/10 + 4/10 + 1/10 + 1/10 + 0 + 0 = 1.0
Critical χ² (df=5, α=0.05) ≈ 11.07
Since 1.0 < 11.07, fail to reject H₀.

Key Points:

Used for categorical data
Expected frequencies should be ≥5 for each category
Also used for independence tests in contingency tables
Right-tailed test only

Summary Table of Hypothesis Tests

Test	Purpose	Assumptions	Test Statistic
Z-test (single mean)	Compare sample mean to population mean	σ known, n ≥ 30 or normal population	Z = (x̄ - μ₀)/(σ/√n)
Z-test (two means)	Compare two population means	σ₁, σ₂ known, independent samples	Z = (x̄₁ - x̄₂)/√(σ₁²/n₁ + σ₂²/n₂)
t-test (single mean)	Compare sample mean to population mean	σ unknown, n < 30, normal population	t = (x̄ - μ₀)/(s/√n), df=n-1
t-test (two means)	Compare two population means	σ unknown, independent samples, equal variances	t = (x̄₁ - x̄₂)/[sₚ√(1/n₁ + 1/n₂)], df=n₁+n₂-2
Paired t-test	Compare means from paired measurements	Differences normally distributed	t = d̄/(s_d/√n), df=n-1
F-test	Compare two variances	Normal populations, independent samples	F = s₁²/s₂², df₁=n₁-1, df₂=n₂-1
Chi-square	Goodness of fit or independence	Categorical data, expected frequencies ≥5	χ² = Σ[(Oᵢ - Eᵢ)²/Eᵢ]

Decision Making in Hypothesis Testing

p-value approach: Compare p-value (probability of observing test statistic under H₀) with α.

If p-value ≤ α, reject H₀
If p-value > α, fail to reject H₀

Critical value approach: Compare test statistic with critical value from distribution table.

If test statistic falls in rejection region, reject H₀
Otherwise, fail to reject H₀

Important Notes:

Never "accept" H₀ - we either reject or fail to reject
Statistical significance ≠ practical significance
Effect size measures the magnitude of the difference
Confidence intervals can provide more information than tests

Common Mistakes in Hypothesis Testing

Using wrong test for given data and assumptions
Ignoring assumptions of the test (normality, equal variance, etc.)
Confusing statistical significance with practical importance
Performing multiple tests without adjustment (increases Type I error)
Stopping data collection after achieving significant result (p-hacking)
Misinterpreting p-value as probability H₀ is true
Using one-tailed test when two-tailed is appropriate
Not reporting effect size along with p-value

Exam Tips for Hypothesis Testing Questions

Problem-Solving Strategy:

Identify the type of problem (mean, proportion, variance, etc.)
Check given information (sample size, known/unknown variance)
Formulate H₀ and H₁ correctly
Choose appropriate test based on conditions
Verify test assumptions
Calculate test statistic carefully
Find critical value or p-value
Make decision and state conclusion in context

Common Exam Questions:

Identify type of error (Type I/II) in given scenario
Interpret p-value in context
Calculate power of test
Determine required sample size for given power
Choose appropriate test for given research question
Interpret confidence interval in relation to hypothesis test

Practice Problems

Problem 1: A sample of 25 students has mean IQ of 110. Population mean is 100 with σ=15. Test if sample is smarter at α=0.01.

Solution: Z-test (σ known)
Z = (110-100)/(15/5) = 3.33
Critical Z (one-tailed) = 2.326
Reject H₀, sample is significantly smarter.

Problem 2: Two machines produce components. Machine A (n=10, s=0.8), Machine B (n=12, s=0.5). Test if variances differ at α=0.05.

Solution: F-test
F = (0.8)²/(0.5)² = 0.64/0.25 = 2.56
Critical F (df₁=9, df₂=11) ≈ 2.90
Fail to reject H₀, no significant difference in variances.