MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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![### Educational Resource: Analyzing Vocabulary Development in Children Based on Age
#### Presented Data:
The accompanying table shows the ages (in years) of 11 children and the number of words in their vocabulary. Below, we will complete the following tasks:
- Display the data in a scatter plot.
- Calculate the sample correlation coefficient \( r \).
#### Data Table:
| Age (years), \( x \) | Vocabulary Size (words), \( y \) |
|-------------|---------------------------|
| 1 | 9 |
| 2 | 240 |
| 3 | 540 |
| 4 | 1100 |
| 5 | 2300 |
| 6 | 2600 |
| 3 | 650 |
| 5 | 2200 |
| 2 | 270 |
| 4 | 1300 |
| 6 | 2300 |
#### (a) Display the Data on a Scatter Plot:
Choosing the correct scatter plot is important for visualizing the data accurately. Among the four scatter plots provided (A, B, C, D), each contains a graphical representation of the given data with Age (years) on the \( x \)-axis and Vocabulary Size on the \( y \)-axis. For clarity, refer to the patterns of data points in the scatter plot.
**Selected Scatter Plot:**
- **Scatter Plot (Option D):**
- This option accurately represents the distribution of data points based on the given table. The other options do not display the correct arrangement of data points.
#### (b) Calculate the Sample Correlation Coefficient \( r \):
The sample correlation coefficient \( r \) measures the strength and direction of a linear relationship between age and vocabulary size.
To calculate this, follow these steps:
1. Calculate the mean of \( x \) values and \( y \) values.
2. Use the formula for \( r \):
\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)} {\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}
\]
3. Round \( r \) to three decimal places as needed.
#### Additional Resources:
- Click [here](#) to view](https://content.bartleby.com/qna-images/question/84435934-315d-4f3f-bf89-3ba66c6d068b/bbe8cfda-f0b9-44ed-a2c0-74afb446f237/rur7dxu_processed.png)
Transcribed Image Text:### Educational Resource: Analyzing Vocabulary Development in Children Based on Age
#### Presented Data:
The accompanying table shows the ages (in years) of 11 children and the number of words in their vocabulary. Below, we will complete the following tasks:
- Display the data in a scatter plot.
- Calculate the sample correlation coefficient \( r \).
#### Data Table:
| Age (years), \( x \) | Vocabulary Size (words), \( y \) |
|-------------|---------------------------|
| 1 | 9 |
| 2 | 240 |
| 3 | 540 |
| 4 | 1100 |
| 5 | 2300 |
| 6 | 2600 |
| 3 | 650 |
| 5 | 2200 |
| 2 | 270 |
| 4 | 1300 |
| 6 | 2300 |
#### (a) Display the Data on a Scatter Plot:
Choosing the correct scatter plot is important for visualizing the data accurately. Among the four scatter plots provided (A, B, C, D), each contains a graphical representation of the given data with Age (years) on the \( x \)-axis and Vocabulary Size on the \( y \)-axis. For clarity, refer to the patterns of data points in the scatter plot.
**Selected Scatter Plot:**
- **Scatter Plot (Option D):**
- This option accurately represents the distribution of data points based on the given table. The other options do not display the correct arrangement of data points.
#### (b) Calculate the Sample Correlation Coefficient \( r \):
The sample correlation coefficient \( r \) measures the strength and direction of a linear relationship between age and vocabulary size.
To calculate this, follow these steps:
1. Calculate the mean of \( x \) values and \( y \) values.
2. Use the formula for \( r \):
\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)} {\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}
\]
3. Round \( r \) to three decimal places as needed.
#### Additional Resources:
- Click [here](#) to view
![### Analyzing the Relationship Between Age and Vocabulary Size in Children
The accompanying table displays the ages (in years) of 11 children alongside the number of words in their vocabulary. This exercise includes plotting the data on a scatter plot and calculating the sample correlation coefficient. Follow the instructions below to understand the relationship between age and vocabulary size.
#### Data Visualization
(a) **Display the data in a scatter plot. Choose the correct graph below.**
Four scatter plots (A, B, C, D) are provided as options:
- **Graph A** is indicated as the correct choice.
- **Graph A** plots `Age (years)` on the x-axis (0 to 8 years) and `Vocabulary Size` on the y-axis (0 to 3000 words). The data points seem to show an upward trend suggesting that as age increases, vocabulary size also increases.
- **Graph B and C** both plot variables with ranges from 0 to 3000 on both x and y axes but are incorrect, likely because of the misrepresentation of data scaling.
- **Graph D** similarly plots `Age (years)` on the x-axis (0 to 8 years) and `Vocabulary Size` on the y-axis (0 to 3000 words), but the scaling or positioning of data points are incorrect.
#### Statistical Analysis
(b) **Calculate the sample correlation coefficient \( r \).**
- The correlation coefficient \( r \) measures the strength and direction of the linear relationship between age and vocabulary size.
- Use the appropriate formula or computational tool to determine \( r \) and round to three decimal places as needed.
Input Box for \( r \): [ ]
To complete this exercise, ensure to confirm the accurate calculation of \( r \), which quantitatively indicates the relationship trends observed in the scatter plot.
---
For teachers and students engaging with this material:
- **Scatter plots** help in visualizing data correlations.
- **Correlation coefficient \( r \)** provides a numerical measure for data analysis.
Feel free to interact with the data tables provided and verify critical correlation values using the provided links. This hands-on approach helps in grasping fundamental statistical concepts.](https://content.bartleby.com/qna-images/question/84435934-315d-4f3f-bf89-3ba66c6d068b/bbe8cfda-f0b9-44ed-a2c0-74afb446f237/z166dy_processed.png)
Transcribed Image Text:### Analyzing the Relationship Between Age and Vocabulary Size in Children
The accompanying table displays the ages (in years) of 11 children alongside the number of words in their vocabulary. This exercise includes plotting the data on a scatter plot and calculating the sample correlation coefficient. Follow the instructions below to understand the relationship between age and vocabulary size.
#### Data Visualization
(a) **Display the data in a scatter plot. Choose the correct graph below.**
Four scatter plots (A, B, C, D) are provided as options:
- **Graph A** is indicated as the correct choice.
- **Graph A** plots `Age (years)` on the x-axis (0 to 8 years) and `Vocabulary Size` on the y-axis (0 to 3000 words). The data points seem to show an upward trend suggesting that as age increases, vocabulary size also increases.
- **Graph B and C** both plot variables with ranges from 0 to 3000 on both x and y axes but are incorrect, likely because of the misrepresentation of data scaling.
- **Graph D** similarly plots `Age (years)` on the x-axis (0 to 8 years) and `Vocabulary Size` on the y-axis (0 to 3000 words), but the scaling or positioning of data points are incorrect.
#### Statistical Analysis
(b) **Calculate the sample correlation coefficient \( r \).**
- The correlation coefficient \( r \) measures the strength and direction of the linear relationship between age and vocabulary size.
- Use the appropriate formula or computational tool to determine \( r \) and round to three decimal places as needed.
Input Box for \( r \): [ ]
To complete this exercise, ensure to confirm the accurate calculation of \( r \), which quantitatively indicates the relationship trends observed in the scatter plot.
---
For teachers and students engaging with this material:
- **Scatter plots** help in visualizing data correlations.
- **Correlation coefficient \( r \)** provides a numerical measure for data analysis.
Feel free to interact with the data tables provided and verify critical correlation values using the provided links. This hands-on approach helps in grasping fundamental statistical concepts.
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