---
title: "Estimation; good estimator; CI for mean"
subject: "Probability and Statistics"
chapter: 6
key: "semester_5/probability_and_statistics:estimation-good-estimator-ci-400-students"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/estimation-good-estimator-ci-400-students
site: BCT Engineering
---

# Estimation; good estimator; CI for mean

Probability and Statistics · Chapter 6: Inference concerning mean

- Asked in: 2075 Chaitra, 6 marks
- Video: [Point Estimate for a Mean and Confidence Interval](https://www.youtube.com/watch?v=6BjHV3KLqdU) — sturmmath
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/estimation-good-estimator-ci-400-students
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-6

## Question

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.

## 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).

## Answer

#### 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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which is a point estimate?
   A. A range for μ
   B. x̄ as a single value for μ
   C. A hypothesis
   D. A confidence level
   Answer: B. A point estimate is one value; interval estimation gives a range. (Key point: Estimation)

2. What is unbiasedness?
   A. Variance tends to 0
   B. Smallest variance
   C. E(estimator) = parameter
   D. Uses all sample information
   Answer: C. An unbiased estimator has expected value equal to the parameter. (Key point: Unbiasedness)

3. Standard error of the mean?
   A. σ/√n = 0.50
   B. σ·√n
   C. σ/n
   D. σ²/n = 0.25
   Answer: A. 10/√400 = 0.5. (Key point: Standard error)

4. 95% CI for μ?
   A. (55.02, 56.98)
   B. (54.71, 57.29)
   C. (55.5, 56.5)
   D. (46.2, 65.8)
   Answer: A. 56 ± 1.96(0.5) = 56 ± 0.98. (Key point: 95% CI)

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