pectancy in years of a given country (x). e results of the regression were: y=a+bx a=-0.887 b-0.055 a) Write the equation of the Least Squares Regression line of the form +
pectancy in years of a given country (x). e results of the regression were: y=a+bx a=-0.887 b-0.055 a) Write the equation of the Least Squares Regression line of the form +
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![### Understanding the Relationship Between Happiness Index and Life Expectancy
To explore the correlation between the happiness index (y) and life expectancy in years (x) of various countries, a regression analysis was conducted. The results of the regression analysis are as follows:
The regression equation is given by:
\[
\hat{y} = a + bx
\]
Where,
- \(a = -0.887\)
- \(b = 0.055\)
#### Questions
**(a) Write the equation of the Least Squares Regression line of the form: \(\hat{y} = a + bx\)**
Answer:
\[
\hat{y} = -0.887 + 0.055x
\]
**(b) Which is a possible value for the correlation coefficient, \(r\)?**
Options:
- 0.749
- -1.43
- 1.43
- -0.749
**Answer:**
The possible value for the correlation coefficient, \(r\), must fall within the range of -1 and 1. Therefore, the choices are:
- 0.749
- -0.749
**(c) If a country increases its life expectancy, the happiness index will…**
Options:
- increase
- decrease
**Answer:**
Increase
### Discussion
The positive slope (\(b = 0.055\)) in the regression equation indicates a positive relationship between life expectancy and the happiness index, suggesting that as life expectancy increases, the happiness index also tends to increase. The plausible values for the correlation coefficient, given in the options, reflect the strength and direction of this relationship. As a general rule, only values between -1 and 1 are valid for correlation coefficients.
This analysis helps in understanding how improvements in life expectancy could potentially enhance overall happiness, which is crucial for policy-making and developmental strategies.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F44029bcf-9042-4df6-b542-d8d45dd97a34%2F053dcb43-686e-4488-a419-1b3e21131b85%2Fm2hasyc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding the Relationship Between Happiness Index and Life Expectancy
To explore the correlation between the happiness index (y) and life expectancy in years (x) of various countries, a regression analysis was conducted. The results of the regression analysis are as follows:
The regression equation is given by:
\[
\hat{y} = a + bx
\]
Where,
- \(a = -0.887\)
- \(b = 0.055\)
#### Questions
**(a) Write the equation of the Least Squares Regression line of the form: \(\hat{y} = a + bx\)**
Answer:
\[
\hat{y} = -0.887 + 0.055x
\]
**(b) Which is a possible value for the correlation coefficient, \(r\)?**
Options:
- 0.749
- -1.43
- 1.43
- -0.749
**Answer:**
The possible value for the correlation coefficient, \(r\), must fall within the range of -1 and 1. Therefore, the choices are:
- 0.749
- -0.749
**(c) If a country increases its life expectancy, the happiness index will…**
Options:
- increase
- decrease
**Answer:**
Increase
### Discussion
The positive slope (\(b = 0.055\)) in the regression equation indicates a positive relationship between life expectancy and the happiness index, suggesting that as life expectancy increases, the happiness index also tends to increase. The plausible values for the correlation coefficient, given in the options, reflect the strength and direction of this relationship. As a general rule, only values between -1 and 1 are valid for correlation coefficients.
This analysis helps in understanding how improvements in life expectancy could potentially enhance overall happiness, which is crucial for policy-making and developmental strategies.
![### Regression Analysis Assignment
#### Question (d)
If the life expectancy is increased by 0.5 years in a certain country, how much will the happiness index change? Round to two decimal places.
[Input box for student response]
#### Question (e)
Use the regression line to predict the happiness index of a country with a life expectancy of 79 years. Round to two decimal places.
[Input box for student response]
---
**Explanation for Students:**
In this part of the assignment, your task is to utilize a given regression line to answer two questions related to life expectancy and the happiness index.
1. **Question (d)** requires you to find out how a small change (an increase of 0.5 years) in life expectancy will affect the happiness index.
2. **Question (e)** involves predicting the happiness index for a specific life expectancy value (79 years) using the regression line.
Please ensure your answers are rounded to two decimal places as instructed. Input your responses in the provided boxes.
---
For further guidance on using regression lines and applying them to real-world data, refer to the supplementary materials provided on the website.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F44029bcf-9042-4df6-b542-d8d45dd97a34%2F053dcb43-686e-4488-a419-1b3e21131b85%2Fjei6wpc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Regression Analysis Assignment
#### Question (d)
If the life expectancy is increased by 0.5 years in a certain country, how much will the happiness index change? Round to two decimal places.
[Input box for student response]
#### Question (e)
Use the regression line to predict the happiness index of a country with a life expectancy of 79 years. Round to two decimal places.
[Input box for student response]
---
**Explanation for Students:**
In this part of the assignment, your task is to utilize a given regression line to answer two questions related to life expectancy and the happiness index.
1. **Question (d)** requires you to find out how a small change (an increase of 0.5 years) in life expectancy will affect the happiness index.
2. **Question (e)** involves predicting the happiness index for a specific life expectancy value (79 years) using the regression line.
Please ensure your answers are rounded to two decimal places as instructed. Input your responses in the provided boxes.
---
For further guidance on using regression lines and applying them to real-world data, refer to the supplementary materials provided on the website.
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