IIT JAM - Mathematical Statistics (MS) · India · IITs (JAM 2026 organised by IIT Bombay)
Original practice questions written to the published syllabus · not past papersMonitoring and accepting quality: Shewhart charts for variables and attributes, CUSUM and EWMA charts, process capability, acceptance sampling plans with their OC, AOQ and ATI curves, and reliability models, applied to seed testing, food processing and input quality. The Agricultural Engineering track covers control charts briefly; this is the full subject.
In the exams: UPSC ISS Statistics III (heavy, reported); SSC CGL JSO (present); AICE-JRF Dairy Microbiology syllabus names 'statistical quality control' (present); light in AICE-JRF Ag Stat
How our questions for IIT JAM spread across the units of Statistical Quality Control & Reliability. Difficulty: 6 easy, 25 medium, 5 hard.
| CUSUM and EWMA charts | 6 | |
| Control charts for variables: X-bar, R and s charts | 5 | |
| Acceptance sampling by variables | 5 | |
| Reliability: hazard rate, series and parallel systems, life testing | 5 | |
| Six Sigma and design for quality (introduction) | 4 | |
| Chance and assignable causes of variation | 3 | |
| Process capability indices | 3 | |
| Control charts for attributes: p, np, c and u charts | 2 | |
| Acceptance sampling by attributes: single, double and sequential plans | 2 | |
| OC, AOQ, AOQL, ASN and ATI curves | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
With R̄ = 0.40 kg from subgroups of 5 (d₂ = 2.326), what is σ̂?
B. 0.172 kg
σ̂ = R̄/d₂ = 0.40/2.326 = 0.172.
Further reading: Montgomery, Introduction to Statistical Quality Control; Duncan, Quality Control and Industrial Statistics; Schilling and Neubauer, Acceptance Sampling in Quality Control
Only one moisture reading per batch is available. Which chart is used?
A. Individuals and moving-range charts
With n = 1 no within-subgroup range exists; the moving range of successive values estimates σ (MR̄/1.128).
Further reading: Montgomery, Introduction to Statistical Quality Control; Duncan, Quality Control and Industrial Statistics; Schilling and Neubauer, Acceptance Sampling in Quality Control
An X̄ chart uses subgroups of 5 with 3-sigma limits. The mean shifts by one process standard deviation. What is the probability that the next point signals?
B. 0.222
The shift is δ√n = √5 = 2.236 standard errors: P(Z > 3 − 2.236) + P(Z < −3 − 2.236) = 0.222; ARL₁ ≈ 4.5. 0.159 ignores √n (a shift of one standard error).
Further reading: Montgomery, Introduction to Statistical Quality Control; Duncan, Quality Control and Industrial Statistics; Schilling and Neubauer, Acceptance Sampling in Quality Control
What is the constant d₂ used in σ̂ = R̄/d₂?
A. Expected range of n standard normals
E(R) = d₂σ for normal samples (2.326 for n = 5), so R̄/d₂ is unbiased for σ.
Further reading: Montgomery, Introduction to Statistical Quality Control; Duncan, Quality Control and Industrial Statistics; Schilling and Neubauer, Acceptance Sampling in Quality Control
With p̄ = 0.04, how large must n be for the p chart to have a positive lower limit?
C. More than 216
LCL > 0 requires p̄ > 3√[p̄(1 − p̄)/n], i.e. n > 9(1 − p̄)/p̄ = 216.
Further reading: Montgomery, Introduction to Statistical Quality Control; Duncan, Quality Control and Industrial Statistics; Schilling and Neubauer, Acceptance Sampling in Quality Control
An EWMA uses λ = 0.2 and L = 3. What are its steady-state limits?
C. μ₀ ± σ
Lσ√[λ/(2 − λ)] = 3σ√(0.2/1.8) = 3σ/3 = σ.
Further reading: Montgomery, Introduction to Statistical Quality Control; Duncan, Quality Control and Industrial Statistics; Schilling and Neubauer, Acceptance Sampling in Quality Control
A dairy laboratory has one fat reading per batch, slightly skewed, and wants to detect slow drifts. Which chart suits it?
C. An EWMA chart
EWMA averaging makes it robust to mild non-normality and sensitive to small drifts with individual data.
Further reading: Montgomery, Introduction to Statistical Quality Control; Duncan, Quality Control and Industrial Statistics; Schilling and Neubauer, Acceptance Sampling in Quality Control
A filler's σ is halved from 8 g to 4 g. Keeping the same protection against under-weight packs, by about how much can the mean fill be lowered with a 3σ margin?
D. 12 g
The margin above the lower limit falls from 3 × 8 = 24 g to 3 × 4 = 12 g, saving 12 g of give-away per pack.
Further reading: Montgomery, Introduction to Statistical Quality Control; Duncan, Quality Control and Industrial Statistics; Schilling and Neubauer, Acceptance Sampling in Quality Control
Which international standard provides variables plans matched to ISO 2859?
B. ISO 3951
ISO 3951 covers inspection by variables; ISO 9001 is a management-system standard.
Further reading: Montgomery, Introduction to Statistical Quality Control; Duncan, Quality Control and Industrial Statistics; Schilling and Neubauer, Acceptance Sampling in Quality Control
Three identical sensors (reliability 0.9 each) form a 2-out-of-3 system. What is its reliability?
A. 0.972
P(at least 2 work) = 3(0.9)²(0.1) + 0.9³ = 0.243 + 0.729 = 0.972.
Further reading: Montgomery, Introduction to Statistical Quality Control; Duncan, Quality Control and Industrial Statistics; Schilling and Neubauer, Acceptance Sampling in Quality Control
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