---
title: "Population, parameter, statistic; samples of 3,7,11,15"
subject: "Probability and Statistics"
chapter: 4
key: "semester_5/probability_and_statistics:population-parameter-statistic-3-7-11"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/population-parameter-statistic-3-7-11
site: BCT Engineering
---

# Population, parameter, statistic; samples of 3,7,11,15

Probability and Statistics · Chapter 4: Sampling Distribution

- Asked in: 2074 Asoj, 6 marks
- Video: [Statistic vs Parameter & Population vs Sample](https://www.youtube.com/watch?v=Mb9BuEkbaHQ) — The Organic Chemistry Tutor
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/population-parameter-statistic-3-7-11
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-4

## Question

Define population. Sample parameter and statistic with suitable examples. A population consists of 3, 7, 11, 15. Consider all possible samples of size two which can be drawn without replacement from this population. Find population mean and Standard error of mean.

## Short answer

Population is the complete set under study; a parameter is a numerical measure of the population, and a statistic is a numerical measure from a sample. For 3,7,11,15 with samples of size 2 without replacement, population mean μ=9 and SE=(4.4721/√2)√(2/3)=2.5820.

## Answer

#### Definitions
- **Population:** the complete collection of all individuals or items under study, for example all 2000 IOE students of a campus.
- **Sample:** a part of the population selected for study, for example 100 of those students.
- **Parameter:** a numerical measure of a population, such as the population mean μ, standard deviation σ or proportion P. It is fixed (usually unknown).
- **Statistic:** a numerical measure computed from a sample, such as the sample mean x̄, standard deviation s or proportion p̂. It varies from sample to sample.

Example: the average marks of all 2000 students is a parameter (μ); the average of the 100 sampled students is a statistic (x̄).

#### Numerical
Population: 3, 7, 11, 15 (N = 4), n = 2 without replacement, 6 samples.

**Population mean and standard deviation**

μ = Σx/N = 36/4 = **9**

σ² = Σ(x − μ)²/N = (36 + 4 + 4 + 36)/4 = 80/4 = 20

σ = √20 = **4.4721**

**All samples**

| No. | Sample | Sample mean |
|---|---|---|
| 1 | (3, 7) | 5 |
| 2 | (3, 11) | 7 |
| 3 | (3, 15) | 9 |
| 4 | (7, 11) | 9 |
| 5 | (7, 15) | 11 |
| 6 | (11, 15) | 13 |

Sum of sample means = 5 + 7 + 9 + 9 + 11 + 13 = 54

Mean of sample means = 54/6 = 9

(equal to μ = 9).

**Standard error of the mean**

With sampling without replacement from a finite population:

SE = (σ/√n) × √((N − n)/(N − 1)) = (4.4721/√2) × √((4 − 2)/(4 − 1))

= 3.1623 × 0.8165 = **2.5820**

(Direct check: the standard deviation of the 6 sample means is 2.5820.)

**Answer: population mean = 9; standard error of the mean = 2.5820.**

## Check yourself

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

1. Which is a parameter?
   A. Average marks of all 2000 students
   B. Average marks of 100 sampled students
   C. Sample proportion p̂
   D. Sample standard deviation s
   Answer: A. A parameter describes the whole population and is fixed; a statistic comes from a sample and varies. (Key point: Parameter vs statistic)

2. For 3, 7, 11, 15, μ and σ are:
   A. 9 and sqrt(20) = 4.472
   B. 9 and 20
   C. 10 and 4.472
   D. 36 and 4.472
   Answer: A. μ = 36/4 = 9 and σ^2 = 80/4 = 20. (Key point: Population mean and SD)

3. The SE of the mean (n = 2, without replacement, N = 4) is:
   A. 3.1623
   B. 2.5820
   C. 4.4721
   D. 2.2361
   Answer: B. (4.4721/√2) × sqrt(2/3) = 3.1623 × 0.8165 = 2.5820. (Key point: Standard error)

4. The six sample means total 54, so their mean is:
   A. 9, equal to μ
   B. 54
   C. 6
   D. 13
   Answer: A. 54/6 = 9 = μ. (Key point: Mean of sample means)

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