--- title: "Point/interval estimation; CI for accident proportion" subject: "Probability and Statistics" chapter: 7 key: "semester_5/probability_and_statistics:point-interval-estimation-accidents-ci" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/point-interval-estimation-accidents-ci site: BCT Engineering --- # Point/interval estimation; CI for accident proportion Probability and Statistics · Chapter 7: Inference concerning proportion - Asked in: 2074 Asoj, 5 marks - Video: [Confidence Intervals for Sample Proportions](https://www.youtube.com/watch?v=W_M1mH5Tqys) — searching4math - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/point-interval-estimation-accidents-ci - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-7 ## Question Explain the concept of (i) point estimation and (ii) Interval estimation of population properties. In a random sample of 400 industrial accidents, it was found that 231 were due to at least partially to unsafe working condition. 95% confidence intervals for the corresponding true proportion. ## Short answer Point estimation uses a single sample statistic (e.g., p̂ = 0.5775) to estimate the parameter. Interval estimation provides a range: p̂ ± z·SE. For n=400, 95% CI is (0.529, 0.626) for the true proportion of accidents partly due to unsafe conditions. ## Answer #### (i) Point estimation A single value computed from the sample is used as the estimate of the unknown population parameter. The sample mean x̄ estimates μ, the sample proportion p̂ estimates P, and s² estimates σ². A good point estimator is unbiased, consistent, efficient and sufficient. It gives no idea of the error involved. #### (ii) Interval estimation An interval (L, U) is computed so that it contains the parameter with a stated probability, the confidence level (1 − α). It is based on the point estimate and its standard error: estimate ± (critical value) × (standard error). For a proportion, the interval is p̂ ± z·√(p̂q̂/n). #### 95% confidence interval for the true proportion ```text n = 400, x = 231 p̂ = 231/400 = 0.5775, q̂ = 0.4225 SE = √(p̂q̂/n) = √(0.5775×0.4225/400) = 0.0247 margin = 1.960 × 0.0247 = 0.0484 limits = 0.5775 ± 0.0484 = (0.5291, 0.6259) ``` **Answer: the 95% confidence interval for the true proportion of accidents partly due to unsafe working conditions is (0.529, 0.626), i.e. about 52.9% to 62.6%.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is point estimation? A. Giving a range that contains the parameter with stated probability B. Using a single sample value as the estimate of an unknown parameter C. Testing a claim about a parameter D. Finding the exact value of the parameter Answer: B. A point estimate is one number computed from the sample, such as x-bar for mu, and it does not show the size of the error. (Key point: Point estimation) 2. What general form does a confidence interval take? A. estimate / standard error B. estimate x critical value C. estimate +/- (critical value) x (standard error) D. estimate +/- sample size Answer: C. Intervals are built as point estimate plus and minus critical value times standard error. (Key point: Interval estimation) 3. For 231 of 400 accidents, what is the standard error of the sample proportion? A. sqrt(0.5775 x 0.4225 / 400) = 0.0247 B. sqrt(0.5775 x 0.4225) = 0.494 C. 0.5775 / 400 = 0.0014 D. sqrt(0.5775 / 400) = 0.038 Answer: A. SE = sqrt(p-hat q-hat / n), with p-hat = 231/400 = 0.5775. (Key point: Sample proportion and SE) 4. Which is the 95% confidence interval for the true proportion? A. (0.553, 0.602) B. (0.529, 0.626) C. (0.481, 0.674) D. (0.5775, 0.626) Answer: B. Margin = 1.96 x 0.0247 = 0.0484, so 0.5775 +/- 0.0484 = (0.529, 0.626). (Key point: 95% confidence interval result) ## Related in Probability and Statistics - [Point vs interval estimation; estimation vs testing](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apoint-vs-interval-estimation-hypothesis-testing) — Probability and Statistics, chapter 6 - [Advantages of interval over point estimation](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aadvantages-interval-estimation-over-point) — Probability and Statistics, chapter 6 - [CI and sample size: overcharged households](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apostal-survey-overcharged-ci-sample-size) — Probability and Statistics, chapter 7 - [Estimation; good estimator; CI for mean](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aestimation-good-estimator-ci-400-students) — Probability and Statistics, chapter 6 - [Estimation vs testing; CI for hardness](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aestimation-vs-hypothesis-ci-magnesium) — Probability and Statistics, chapter 6 - [CI: sample of 16, mean 52, SD 4](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aci-sample-16-mean-52-sd) — Probability and Statistics, chapter 6 ## Related in other subjects - [Actual load estimation in industrial plants](https://www.bctengineer.com/question.md?key=semester_6%2Findustrial_power_distribution_and_illumination%3Aindustrial-load-estimation-method) — Industrial Power Distribution and Illumination, chapter 1, asked 2× - [Random number properties and Gap test](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Arn-properties-gap-test) — Simulation and Modeling, chapter 6, asked 4× - [Solid breakdown due to actual working conditions](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Asolid-breakdown-working-conditions) — High Voltage Engineering, chapter 6 - [Random number properties; KS test algorithm](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Arn-properties-ks-algorithm) — Simulation and Modeling, chapter 6