---
title: "Sampling distribution: population 2,4,6,8"
subject: "Probability and Statistics"
chapter: 4
key: "semester_5/probability_and_statistics:population-2-4-6-8-samples"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/population-2-4-6-8-samples
site: BCT Engineering
---

# Sampling distribution: population 2,4,6,8

Probability and Statistics · Chapter 4: Sampling Distribution

- Asked in: 2074 Chaitra, 4 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-2-4-6-8-samples
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-4

## Question

A population consists of five numbers 2, 4, 6 and 8
a) Enumerate all possible sample of size two without replacement
b) Show that the mean of the sampling distribution of sample mean is equal to population mean

## Short answer

Using N=4 population 2,4,6,8, all six samples of size two without replacement have means 3,4,5,5,6,7. Population mean is 5 and mean of sample means is 30/6=5, so the equality is verified.

## Answer

The population has four numbers: 2, 4, 6, 8 (N = 4; the question says "five" but lists four, so N = 4 is used). n = 2, without replacement, so there are ⁴C₂ = 6 samples.

#### a) All possible samples of size two
| No. | Sample | Sample mean |
|---|---|---|
| 1 | (2, 4) | 3 |
| 2 | (2, 6) | 4 |
| 3 | (2, 8) | 5 |
| 4 | (4, 6) | 5 |
| 5 | (4, 8) | 6 |
| 6 | (6, 8) | 7 |

#### b) Mean of sample means equals population mean
Population mean: μ = (2 + 4 + 6 + 8)/4 = 20/4 = 5

Sum of sample means = 3 + 4 + 5 + 5 + 6 + 7 = 30

Mean of sample means = 30/6 = 5

Since the mean of the sampling distribution of x̄ = 5 = μ = 5, the statement is shown to be true.

## Check yourself

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

1. How many samples of size 2 without replacement come from 2, 4, 6, 8?
   A. 16
   B. 6
   C. 8
   D. 12
   Answer: B. 4C2 = 6. (Key point: Number of samples)

2. The population mean of 2, 4, 6, 8 is:
   A. 4
   B. 5
   C. 6
   D. 20
   Answer: B. 20/4 = 5. (Key point: Population mean)

3. The six sample means are 3, 4, 5, 5, 6, 7; their mean is:
   A. 5, equal to μ
   B. 30
   C. 4.5
   D. 6
   Answer: A. 30/6 = 5, which equals the population mean. (Key point: Sample means sum)

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