---
title: "Sampling distribution: population 2,8,14,20"
subject: "Probability and Statistics"
chapter: 4
key: "semester_5/probability_and_statistics:population-2-8-14-20-samples"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/population-2-8-14-20-samples
site: BCT Engineering
---

# Sampling distribution: population 2,8,14,20

Probability and Statistics · Chapter 4: Sampling Distribution

- Asked in: 2075 Asoj, 6 marks
- Video: [COSM - Sampling Distribution of Means Without Replacement](https://www.youtube.com/watch?v=q-bw5j0RcJs) — Vardhaman College of Engineering
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/population-2-8-14-20-samples
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-4

## Question

A population consists of four number 2, 8, 14, 20.
a) Write down all possible sample size of two without replacement.
b) Verify that the population mean is equal to the mean of the sample mean.

## Short answer

Population 2,8,14,20 has mean 11. The six possible samples of size two without replacement have means 5,8,11,11,14,17, whose mean is 66/6=11, equal to the population mean.

## Answer

Population: 2, 8, 14, 20 (N = 4), n = 2, without replacement, number of samples = ⁴C₂ = 6.

#### a) All possible samples
| No. | Sample | Sample mean |
|---|---|---|
| 1 | (2, 8) | 5 |
| 2 | (2, 14) | 8 |
| 3 | (2, 20) | 11 |
| 4 | (8, 14) | 11 |
| 5 | (8, 20) | 14 |
| 6 | (14, 20) | 17 |

#### b) Population mean equals mean of sample means
Population mean: μ = (2 + 8 + 14 + 20)/4 = 44/4 = 11

Sum of sample means = 5 + 8 + 11 + 11 + 14 + 17 = 66

Mean of sample means = 66/6 = 11

Hence mean of the sample means (11) = population mean (11). Verified.

## Check yourself

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

1. Samples of size 2 without replacement from 4 numbers number:
   A. 6
   B. 8
   C. 16
   D. 4
   Answer: A. 4C2 = 6. (Key point: Number of samples)

2. The population mean of 2, 8, 14, 20 is:
   A. 10
   B. 11
   C. 12
   D. 44
   Answer: B. 44/4 = 11. (Key point: Population mean)

3. The sample means are 5, 8, 11, 11, 14, 17. Their mean is:
   A. 11, which equals μ
   B. 66
   C. 10
   D. 12
   Answer: A. 66/6 = 11 = μ, verifying that the mean of x̄ equals the population mean. (Key point: Verification)

## Close variants set in other exams

- [Samples of 2,8,14,20 with standard error](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apopulation-2-8-14-20-with) — Probability and Statistics, chapter 4, asked 2×

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