Module V: Statistical Techniques III – Complete Formulas with Clear Explanation
1. Sampling Theory
| Type | Condition | Standard Error (SE) Formulas |
|---|---|---|
| Large Sample (n ≥ 30 or np, nq > 5) | Population variance known/unknown | SE_ˆp = √[p(1–p)/n] (proportion) SE_x̄ = σ/√n (mean, σ known) SE_x̄ = s/√n (mean, σ unknown) |
| Small Sample (n < 30) | Population normal, σ unknown | Use t-distribution instead of Z |
2. Testing of Hypothesis – Basic Terminology
| Term | Definition / Formula |
|---|---|
| Null Hypothesis (H₀) | Statement of no difference (e.g., μ = μ₀, p = p₀) |
| Alternative Hypothesis (H₁) | Statement of difference (μ ≠ μ₀, μ > μ₀, μ < μ₀) |
| Level of Significance (α) | Probability of Type I error (usually 5% or 1%) |
| Critical Region | Values of test statistic that lead to rejection of H₀ |
| Type I Error | Reject H₀ when it is true (probability = α) |
| Type II Error (β) | Accept H₀ when it is false |
| Power of Test | 1 – β |
| Confidence Limits (for mean) | x̄ ± Z_{α/2} (σ/√n) (large sample) x̄ ± t_{α/2} (s/√n) (small sample, df = n–1) |
3. Tests of Significance
| Test | Test Statistic | Critical Region / Decision Rule | Degrees of Freedom (df) |
|---|---|---|---|
| Z-test (large sample mean) | Z = (x̄ – μ₀) / (σ/√n) | |Z| > Z_{α/2} (two-tail) Z > Z_α (right) Z < –Z_α (left) |
— |
| Z-test (proportion) | Z = (p̂ – p₀) / √[p₀(1–p₀)/n] | Same as above | — |
| t-test (single mean, small) | t = (x̄ – μ₀) / (s/√n) | |t| > t_{α/2}, ν=n–1 (two-tail) | n–1 |
| Paired t-test | t = d̄ / (s_d / √n) (d = difference) | Same | n–1 |
| Two independent means (large) | Z = (x̄₁ – x̄₂) / √(σ₁²/n₁ + σ₂²/n₂) | Same as Z | — |
| Two independent means (small, σ equal) | t = (x̄₁ – x̄₂) / [s_p √(1/n₁ + 1/n₂)] s_p² = [(n₁–1)s₁² + (n₂–1)s₂²]/(n₁+n₂–2) |
|t| > t_{α/2}, ν=n₁+n₂–2 | n₁+n₂–2 |
| F-test (equality of variances) | F = s₁² / s₂² (s₁² > s₂²) | F > F_α, ν₁=n₁–1, ν₂=n₂–1 (one-tail) | ν₁=n₁–1, ν₂=n₂–1 |
| Chi-square variance (single) | χ² = (n–1) s² / σ₀² | χ² > χ²_α or χ² < χ²_{1–α} | n–1 |
| Chi-square Goodness of fit | χ² = Σ (O_i – E_i)² / E_i | χ² > χ²_α | k–1–parameters |
| Chi-square Independence (r×c) | χ² = Σ Σ (O_ij – E_ij)² / E_ij E_ij = (Row total × Col total)/N | χ² > χ²_α | (r–1)(c–1) |
4. One-Way Analysis of Variance (ANOVA)
| Source | SS | df | MS | F-ratio |
|---|---|---|---|---|
| Between groups | SSB = Σ n_j (x̄_j – x̄)² | k–1 | MSB = SSB/(k–1) | F = MSB / MSE |
| Within groups | SSE = Σ Σ (x_ij – x̄_j)² | N–k | MSE = SSE/(N–k) | |
| Total | SST = Σ Σ (x_ij – x̄)² | N–1 | — | — |
H₀: All means equal → Accept if F ≤ F_α(k–1, N–k)
5. Statistical Quality Control (SQC) – Control Charts
A. Control Charts for Variables (Measurable data)
| Chart | Purpose | Center Line | Control Limits | Subgroup size |
|---|---|---|---|---|
| X̄-chart | Control process mean | X̄̄ (grand mean) | UCL = X̄̄ + A₂ R̄ LCL = X̄̄ – A₂ R̄ |
n = 2 to 10 |
| R-chart | Control process variability | R̄ | UCL = D₄ R̄ LCL = D₃ R̄ |
n ≤ 10 |
Constants Table (most commonly used)
| n | A₂ | D₃ | D₄ |
|---|---|---|---|
| 2 | 1.880 | 0 | 3.268 |
| 3 | 1.023 | 0 | 2.574 |
| 4 | 0.729 | 0 | 2.282 |
| 5 | 0.577 | 0 | 2.114 |
| 6 | 0.483 | 0 | 2.004 |
| 7 | 0.419 | 0.076 | 1.924 |
B. Control Charts for Attributes (Count data)
| Chart | Purpose | Center Line | Control Limits |
|---|---|---|---|
| p-chart | Fraction defective | p̄ = Total defectives/Total sample | UCL = p̄ + 3√[p̄(1–p̄)/n] LCL = p̄ – 3√[p̄(1–p̄)/n] |
| np-chart | Number defective (fixed n) | np̄ | UCL = np̄ + 3√[np̄(1–p̄)] LCL = np̄ – 3√[np̄(1–p̄)] |
| c-chart | Number of defects per unit | c̄ = Total defects/No. of samples | UCL = c̄ + 3√c̄ LCL = c̄ – 3√c̄ |
Summary of Most Important Formulas
| Topic | Key Formula |
|---|---|
| Z-test (mean) | Z = (x̄ – μ)/(σ/√n) |
| t-test (single mean) | t = (x̄ – μ)/(s/√n) , df = n–1 |
| Two-sample t-test (equal var) | t = (x̄₁ – x̄₂)/[s_p √(1/n₁ + 1/n₂)] |
| Chi-square variance | χ² = (n–1)s²/σ₀² , df = n–1 |
| F-test | F = s₁²/s₂² |
| ANOVA F-ratio | F = MSB/MSE |
| X̄-chart limits | X̄̄ ± A₂ R̄ |
| R-chart limits | D₄ R̄ , D₃ R̄ |
| p-chart limits | p̄ ± 3√[p̄(1–p̄)/n] |
| c-chart limits | c̄ ± 3√c̄ |
These formulas cover 100% of Module V syllabus (Sampling, Hypothesis Testing, ANOVA & SQC) as per most B.E./B.Tech curricula. Practice numericals on t-test, χ²-test, ANOVA table, and construction of X̄–R and p/c charts – these are the most frequently asked in university exams.