MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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### Sampling Distribution: An Educational Guide

#### Understanding the Problem Statement
Let's delve into a statistical problem involving the assets of the four wealthiest individuals in a country. The assets, in billions of dollars, are as follows:
- 39
- 28
- 22
- 13

We are assuming that samples of size \( n = 2 \) are randomly selected with replacement from this set of four values.

#### Constructing the Sampling Distribution
To create the sampling distribution of the sample mean, we must first identify all 16 possible samples and calculate the mean for each. These values are then grouped, and their corresponding probabilities are determined.

#### The Table of Sample Means
Below is the table constructed from the problem:

| \( \bar{x} \) | Probability |   | \( \bar{x} \) | Probability |
|---------------|-------------|---|---------------|-------------|
| 39            |             |   | 25            |             |
| 33.5          |             |   | 22            |             |
| 30.5          |             |   | 20.5          |             |
| 28            |             |   | 17.5          |             |
| 26            |             |   | 13            |             |

**Note:** Fill in the probabilities as integers or fractions as specified in the problem.

#### Explanation
Here is a detailed explanation of what is expected in each step of the process:

1. **Identify Possible Samples:**
   Each sample is of size 2 and is selected with replacement. This results in a total of \( 4 \times 4 = 16 \) possible samples since each of the four assets can pair with any other, including itself.
   
2. **Calculate the Mean:**
   For each sample, compute the sample mean (\( \bar{x} \)). For example, if the sample is (39, 22), the mean is \( \frac{39 + 22}{2} = 30.5 \).

3. **Count Occurrences:**
   Determine how frequently each unique sample mean occurs among all possible samples.

4. **Determine Probabilities:**
   The probability of each unique sample mean is derived by dividing its frequency by the total number of samples (16).

This method provides a comprehensive approach to understanding how sample means distribute when drawn from a small, finite population. By working these steps, one gains
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Transcribed Image Text:### Sampling Distribution: An Educational Guide #### Understanding the Problem Statement Let's delve into a statistical problem involving the assets of the four wealthiest individuals in a country. The assets, in billions of dollars, are as follows: - 39 - 28 - 22 - 13 We are assuming that samples of size \( n = 2 \) are randomly selected with replacement from this set of four values. #### Constructing the Sampling Distribution To create the sampling distribution of the sample mean, we must first identify all 16 possible samples and calculate the mean for each. These values are then grouped, and their corresponding probabilities are determined. #### The Table of Sample Means Below is the table constructed from the problem: | \( \bar{x} \) | Probability | | \( \bar{x} \) | Probability | |---------------|-------------|---|---------------|-------------| | 39 | | | 25 | | | 33.5 | | | 22 | | | 30.5 | | | 20.5 | | | 28 | | | 17.5 | | | 26 | | | 13 | | **Note:** Fill in the probabilities as integers or fractions as specified in the problem. #### Explanation Here is a detailed explanation of what is expected in each step of the process: 1. **Identify Possible Samples:** Each sample is of size 2 and is selected with replacement. This results in a total of \( 4 \times 4 = 16 \) possible samples since each of the four assets can pair with any other, including itself. 2. **Calculate the Mean:** For each sample, compute the sample mean (\( \bar{x} \)). For example, if the sample is (39, 22), the mean is \( \frac{39 + 22}{2} = 30.5 \). 3. **Count Occurrences:** Determine how frequently each unique sample mean occurs among all possible samples. 4. **Determine Probabilities:** The probability of each unique sample mean is derived by dividing its frequency by the total number of samples (16). This method provides a comprehensive approach to understanding how sample means distribute when drawn from a small, finite population. By working these steps, one gains
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