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Estimation; good estimator; CI for mean

Short answer

Estimation uses sample data to approximate population parameters; a good estimator is unbiased, consistent, efficient, and sufficient. With n = 400, x̄ = 56, σ = 10, 95% CI is (55.02, 56.98) and 99% CI is (54.71, 57.29).

Define estimation? Write characteristics of a good estimator? A sample of 400 students taking Entrance for BE revealed an average score of 56. Construct a 95% as well as 99% confidence interval for population mean score if standard deviation of score of all students is known to be 10.

Estimation

Estimation is the process of using sample data to find approximate values of unknown population parameters. It is of two types: point estimation (a single value, e.g. x̄ for μ) and interval estimation (a range of values, e.g. a confidence interval, within which the parameter is expected to lie with a stated confidence).

Characteristics of a good estimator

  1. Unbiasedness: E(estimator) = parameter. Example: E(x̄) = μ, E(s²) = σ².
  2. Consistency: the estimator approaches the true parameter as the sample size increases (its variance tends to 0).
  3. Efficiency: among unbiased estimators, it has the smallest variance.
  4. Sufficiency: it uses all the information about the parameter contained in the sample.

Confidence intervals for the mean

Given n = 400, x̄ = 56, σ = 10 (known). Standard error σ/√n = 10/√400 = 0.50.

CI: x̄ ± Z_α/2 · σ/√n

95% CI: Z = 1.96. Margin = 1.96 × 0.50 = 0.98 x̄ ± 0.98 = (55.02, 56.98)

99% CI: Z = 2.576. Margin = 2.576 × 0.50 = 1.288 x̄ ± 1.288 = (54.71, 57.29)

Answer: 95% CI = (55.02, 56.98); 99% CI = (54.71, 57.29). The 99% interval is wider because higher confidence needs a larger margin.

✓ Answer checked by Claude Sonnet 5.5

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