- 2074 Chaitra · 8 marks
IOE entrance scores: estimator and consistency
Short answer
Average score: 50.13. Unbiased standard deviation: 7.33, standard error: 0.946. The sample mean is consistent because it is unbiased and its variance σ²/n → 0 as n grows.
Following data reveals the scores of sixty candidates of IOE entrance examination 51.43 40 78.57 46.43 51.43 50.71 42.14 50.71 42.86 55 71.43 64.29 52.86 42.14 57.14 45.71 43.57 40 44.29 55.71 40 48.57 48.57 49.29 51.43 47.14 54.29 45 53.57 50 49.29 60 48.57 50.71 50 49.29 47.14 53.57 58.57 43.57 47.14 53.57 47.86 47.14 40 43.57 52.86 47.86 49.29 49.29 42.86 47.14 48.57 50 47.14 50.71 52.86 47.86 47.14 70 a) Estimate average score of candidates. b) Find unbiased estimator of true standard deviation and standard error of average score. c) Also test for consistency of score.
n = 60 scores. Sum of scores Σx = 3007.85, Σx² = 153956.05.
a) Estimate of the average score
The sample mean is the point estimator of the population mean μ:
x̄ = Σx/n = 3007.85/60 = 50.131
b) Unbiased estimator of σ and standard error of the mean
The sample variance with divisor (n − 1) is an unbiased estimator of σ²:
s² = [Σx² − n·x̄²] / (n − 1) = [153956.05 − 60×2513.100]/59 = 53.729
s = √53.729 = 7.330 (estimate of σ; E(s²) = σ²)
Standard error of the mean: SE(x̄) = s/√n = 7.330/√60 = 7.330/7.746 = 0.946
c) Consistency
An estimator T is consistent for θ if T → θ in probability as n → ∞; sufficient conditions are E(T) → θ and Var(T) → 0.
For the sample mean:
- E(x̄) = μ (unbiased)
- Var(x̄) = σ²/n → 0 as n → ∞
So x̄ is a consistent estimator of μ. Using s = 7.330 as the estimate of σ, the standard error falls as the sample grows:
| n | 15 | 30 | 60 | 240 |
|---|---|---|---|---|
| SE = s/√n | 1.893 | 1.338 | 0.946 | 0.473 |
The spread of x̄ shrinks toward zero, so the average score of the sample becomes a more and more accurate estimate of the true mean; likewise s² is a consistent estimator of σ².
Answer: x̄ = 50.13, s = 7.33, SE = 0.946; x̄ is unbiased and consistent.
✓ Answer checked by Claude Sonnet 5.5
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