Now say a simple linear regression model is fit to the data. Fitting the simple linear regression model, the estimated regression equation is: Ỹi = 6.4285 +1.0534X; or distance; = 6.4285 +1.0534height; d. What is the predicted length of the jump for an athlete who is 72 inches tall? e. Interpret what the 1.0534 represents. f. Does the intercept of 6.4285 inches have any useful interpretation to the coach?
Now say a simple linear regression model is fit to the data. Fitting the simple linear regression model, the estimated regression equation is: Ỹi = 6.4285 +1.0534X; or distance; = 6.4285 +1.0534height; d. What is the predicted length of the jump for an athlete who is 72 inches tall? e. Interpret what the 1.0534 represents. f. Does the intercept of 6.4285 inches have any useful interpretation to the coach?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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**Scatter Plot Explanation:**
- **Axes:**
- The horizontal axis (X-axis) represents the height in inches of the athletes.
- The vertical axis (Y-axis) represents the long jump distance in inches.
- **Data Points:**
- Each point on the scatter plot represents an individual athlete’s height and corresponding jump distance.
- **Range:**
- Heights range from approximately 68 to 78 inches.
- Jump distances range from about 78 to 88 inches.
**Observations:**
- As the height increases, there is an observable trend where the jump distance tends to increase as well.
- The plot shows some variability, indicating that while there is a positive association between height and jump distance, other factors may also influence the jump performance.
This scatter plot is vital for visualizing the correlation between the athlete's height and their long jump capabilities. It can guide coaches in understanding key factors contributing to long jump performance and potentially inform training methods.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b7d2580-6e0c-4034-b41e-8cf2c4a4e82e%2F6b4973e8-26ad-4428-bbea-d96218ff7276%2Fcy4v47n_processed.png&w=3840&q=75)
Transcribed Image Text:### Analyzing the Relationship Between Athlete Height and Long Jump Distance
A high school track & field coach conducted a study to examine the relationship between the height of athletes and their long jump distances. The measurements for both variables are in inches. The height of the athlete serves as the explanatory variable (X), and the jump distance is the response variable (Y).
The data collected from 32 athletes is illustrated in the scatter plot below:
**Scatter Plot Explanation:**
- **Axes:**
- The horizontal axis (X-axis) represents the height in inches of the athletes.
- The vertical axis (Y-axis) represents the long jump distance in inches.
- **Data Points:**
- Each point on the scatter plot represents an individual athlete’s height and corresponding jump distance.
- **Range:**
- Heights range from approximately 68 to 78 inches.
- Jump distances range from about 78 to 88 inches.
**Observations:**
- As the height increases, there is an observable trend where the jump distance tends to increase as well.
- The plot shows some variability, indicating that while there is a positive association between height and jump distance, other factors may also influence the jump performance.
This scatter plot is vital for visualizing the correlation between the athlete's height and their long jump capabilities. It can guide coaches in understanding key factors contributing to long jump performance and potentially inform training methods.
![### Simple Linear Regression Model Example
In our analysis, a simple linear regression model is used to fit the data. Below is the estimated regression equation derived from fitting this simple linear regression model:
\[ \hat{Y}_i = 6.4285 + 1.0534X_i \]
or
\[ \widehat{\text{distance}}_i = 6.4285 + 1.0534 \text{height}_i \]
#### Questions and Interpretations
**d. What is the predicted length of the jump for an athlete who is 72 inches tall?**
To predict the length of the jump for an athlete who is 72 inches tall, we substitute \( \text{height}_i = 72 \) into the regression equation:
\[
\widehat{\text{distance}}_i = 6.4285 + 1.0534 \times 72
\]
Calculate the value:
\[
\widehat{\text{distance}}_i = 6.4285 + 75.8448 = 82.2733
\]
So, the predicted length of the jump for an athlete who is 72 inches tall is 82.2733 inches.
**e. Interpret what the 1.0534 represents.**
The coefficient 1.0534 represents the expected change in the predicted length of the jump for each one-inch increase in the height of the athlete. In other words, for every additional inch in height, the jump length is expected to increase by 1.0534 inches.
**f. Does the intercept of 6.4285 inches have any useful interpretation to the coach?**
The intercept 6.4285 inches represents the predicted length of the jump for an athlete whose height is 0 inches. However, in this context, having a height of 0 inches is not realistic or practical. Therefore, the intercept does not have a meaningful or useful interpretation for the coach in terms of athletic performance. It’s more of a mathematical artifact of the regression model.
This simple linear regression model helps in understanding the relationship between an athlete's height and their jump length, providing insights that can be valuable for training and performance predictions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b7d2580-6e0c-4034-b41e-8cf2c4a4e82e%2F6b4973e8-26ad-4428-bbea-d96218ff7276%2F2lkf5d9_processed.png&w=3840&q=75)
Transcribed Image Text:### Simple Linear Regression Model Example
In our analysis, a simple linear regression model is used to fit the data. Below is the estimated regression equation derived from fitting this simple linear regression model:
\[ \hat{Y}_i = 6.4285 + 1.0534X_i \]
or
\[ \widehat{\text{distance}}_i = 6.4285 + 1.0534 \text{height}_i \]
#### Questions and Interpretations
**d. What is the predicted length of the jump for an athlete who is 72 inches tall?**
To predict the length of the jump for an athlete who is 72 inches tall, we substitute \( \text{height}_i = 72 \) into the regression equation:
\[
\widehat{\text{distance}}_i = 6.4285 + 1.0534 \times 72
\]
Calculate the value:
\[
\widehat{\text{distance}}_i = 6.4285 + 75.8448 = 82.2733
\]
So, the predicted length of the jump for an athlete who is 72 inches tall is 82.2733 inches.
**e. Interpret what the 1.0534 represents.**
The coefficient 1.0534 represents the expected change in the predicted length of the jump for each one-inch increase in the height of the athlete. In other words, for every additional inch in height, the jump length is expected to increase by 1.0534 inches.
**f. Does the intercept of 6.4285 inches have any useful interpretation to the coach?**
The intercept 6.4285 inches represents the predicted length of the jump for an athlete whose height is 0 inches. However, in this context, having a height of 0 inches is not realistic or practical. Therefore, the intercept does not have a meaningful or useful interpretation for the coach in terms of athletic performance. It’s more of a mathematical artifact of the regression model.
This simple linear regression model helps in understanding the relationship between an athlete's height and their jump length, providing insights that can be valuable for training and performance predictions.
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