Problem 9 - Calculus for Biology and Medicine

Question

Assume that a population consists of the three numbers 1, 6 , and 8 . List all samples of size 2 that can be drawn from this population with replacement, and find the sample mean of each sample.

Step-by-Step Solution

Answer

There are 9 samples, with means 1, 3.5, 4.5, 3.5, 6, 7, 4.5, 7, and 8.

Step 1: Understanding the Problem

We're given a population consisting of the numbers 1, 6, and 8, and we need to draw samples of size 2 with replacement. This means after selecting a number, we place it back, so it can be selected again.

Step 2: Listing Possible Samples

Since sampling is with replacement, each sample element can be any of the three available numbers. Thus, there are 9 possible samples: (1,1), (1,6), (1,8), (6,1), (6,6), (6,8), (8,1), (8,6), and (8,8).

Step 3: Calculating Sample Means

For each sample, we calculate the mean by adding the two numbers and dividing by 2. For clarity, \( \text{mean} = \frac{x_1 + x_2}{2} \) where \( x_1 \) and \( x_2 \) are the numbers in the sample.

Step 4: Sample Means Calculation

- Sample (1,1): Mean = \( \frac{1+1}{2} = 1 \)- Sample (1,6): Mean = \( \frac{1+6}{2} = 3.5 \)- Sample (1,8): Mean = \( \frac{1+8}{2} = 4.5 \)- Sample (6,1): Mean = \( \frac{6+1}{2} = 3.5 \)- Sample (6,6): Mean = \( \frac{6+6}{2} = 6 \)- Sample (6,8): Mean = \( \frac{6+8}{2} = 7 \)- Sample (8,1): Mean = \( \frac{8+1}{2} = 4.5 \)- Sample (8,6): Mean = \( \frac{8+6}{2} = 7 \)- Sample (8,8): Mean = \( \frac{8+8}{2} = 8 \)

Key Concepts

Understanding the Population

In statistics, the term "population" refers to the complete set of items or observations that are of interest. In this particular exercise, our population is made up of the numbers 1, 6, and 8. All data points within this population are of equal importance to our study. When dealing with a smaller, finite collection like this, each number in the population can be directly listed and considered individually, which simplifies our calculations.

It is crucial to discern between population and sample data. While the population includes every member of a specified group, a sample consists of merely a part of this group. Here, because the population is only three numbers and very limited, each number plays a significant role in our overall understanding and our final results.

Demystifying Sample Mean

The concept of a sample mean is central to statistics, especially when you're trying to understand characteristics of a sample from a larger population. In our exercise, we are tasked with calculating the sample mean for each group of numbers selected. The sample mean is simply the average of all numbers in the sample.

This is calculated by adding together all numbers in a sample and then dividing the total by the number of observations in the sample. Mathematically, if each sample consists of two numbers, let's denote them as \(x_1\) and \(x_2\), then the sample mean \(\bar{x}\) is found using the formula:

• \( \bar{x} = \frac{x_1 + x_2}{2} \)

Calculating a sample mean helps in understanding the potential center of the data distribution for the sample pairs taken from the population, facilitating comparisons and predictions.

Step-By-Step Solutions for Clarity

A step-by-step solution can greatly assist in making sense of complex problems. Breaking a problem down into manageable parts helps clarify the process and ensures that each part is understood before moving on to the next. In our exercise example, we first identify the problem; namely that our population consists of only three numbers and that samples are taken with replacement.

Next, we list all possible samples. Sampling with replacement means that each selection can occur multiple times. This leads to a specific set of potential samples, in this case, nine pairs. Each pair then becomes a candidate for calculating the sample mean. By methodically applying the mean formula to each sample, we can derive a comprehensive list of means. Understanding these steps empowers you to apply similar processes to other problems, demystifying complex statistical concepts through repetition and clarity.