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UPSC Combined Geo-Scientist

Chemist

  • 3 descriptive papers
  • 200 marks each
  • 3 h per paper

Mains Paper III covers analytical and organic chemistry.

UPSC Combined Geo-Scientist, Mains · 2026 · Mains Paper III

Q3(a). Variance and F-test for two methods of Al in bauxite

  • 10 marks
  • Topic: Statistics in analysis
  • Answered
Question

Aluminium (Al) content (in g) in a 100 g bauxite sample is determined by two different laboratory methods: Method A: 51.0, 51.9, 52.3, 54.4, 50.9, 53.1; Method B: 51.1, 52.0, 52.7, 54.6, 51.2, 53.4. Calculate the `variance' of the analyses by both methods (A and B) above and compare their precision through F-test, and state which of the methods, A or B, is significant for analysis of Al in bauxite ore. [F-table is provided]

Answer in brief

s_A²=1.771, s_B²=1.832; F=1.832/1.771=1.03, far below F_(crit)(5,5)=5.05 (95 %). The two methods are equally precise; there is no significant difference, so either method is suitable (A has marginally smaller variance).

Key points

  • Σ(xᵢ-52.267)²=8.853, divided by 5: s_A²=1.771. (B): Σ(xᵢ-52.500)²=9.160, divided by 5: s_B²=1.832.
  • larger over smaller variance: F=(s_B²)/(s_A²)=(1.832)/(1.771)=1.03
  • Table: φ₁=φ₂=n-1=5: F_(crit)=5.05 (95 % level), 11.0 (99 %).
  • F_(calc)=1.03<5.05: the null hypothesis (σ_A=σ_B) is retained. The precision of the two methods is not significantly different.

The full model answer

Worked out step by step, as it should go in the answer booklet. The solution sheet follows it.

Methodnmean (g)s²=(Σ(xᵢ- x)²)/(n-1)s
A652.271.7711.331
B652.501.8321.354
  1. Variance (A): Σ(xᵢ-52.267)²=8.853, divided by 5: s_A²=1.771. (B): Σ(xᵢ-52.500)²=9.160, divided by 5: s_B²=1.832.
  2. F ratio: larger over smaller variance: F=(s_B²)/(s_A²)=(1.832)/(1.771)=1.03
  3. Table: φ₁=φ₂=n-1=5: F_(crit)=5.05 (95 % level), 11.0 (99 %).
  4. Decision: F_(calc)=1.03<5.05: the null hypothesis (σ_A=σ_B) is retained. The precision of the two methods is not significantly different.
  5. Conclusion: neither method is significantly better; both are acceptable for Al in bauxite. If one must be named, Method A (the smaller variance); the means (52.27 and 52.50 g) can be compared with a t-test.
Model answer to Q3(a), 2026 Mains Paper III: solution page 8
Solution page 8 of 30
Read the whole 2026 solution2026 question paper

Coversvariance, F-testVariance and F-test for two methods of Al in bauxite

All parts of Question 3