To tackle the given problem, we first understand the concept of correlation coefficients. The correlation coefficient is a statistical measure that calculates the strength and direction of a relationship between two variables. Its value ranges between -1 and 1, where:
- 1 indicates a perfect positive correlation;
- -1 indicates a perfect negative correlation; and
- 0 indicates no correlation.
### Data Provided:
1. Happiness Index for 7 years: [36, 36, 34, 32, 36, 33, 30]
2. Response to "Happy with Life" Statement (percentage strongly agree): [82, 83, 78, 83, 80, 82, 79]
3. Response to "Won't Benefit" Statement (percentage somewhat or strongly agree): [31, 37, 35, 39, 37, 42, 39]
### Solution:
#### (a) Correlation between Happiness Index and "Happy with Life" Statement
To calculate the correlation coefficient:
1. Data Points:
- Happiness Index: [36, 36, 34, 32, 36, 33, 30]
- Happy with Life: [82, 83, 78, 83, 80, 82, 79]
2. Calculate Correlation:
[tex]\[
r_{happy} = 0.2493
\][/tex]
The rounded value to four decimal places is 0.2493.
Hence, the correlation coefficient between the Happiness Index and the response to the "Happy with Life" statement is 0.2493.
[tex]\[
\boxed{0.2493}
\][/tex]
#### (b) Correlation between Happiness Index and "Won't Benefit" Statement
To calculate the correlation coefficient:
1. Data Points:
- Happiness Index: [36, 36, 34, 32, 36, 33, 30]
- Won't Benefit: [31, 37, 35, 39, 37, 42, 39]
2. Calculate Correlation:
[tex]\[
r_{wont\_benefit} = -0.5898
\][/tex]
The rounded value to four decimal places is -0.5898.
Hence, the correlation coefficient between the Happiness Index and the response to the "Won't Benefit" statement is -0.5898.
[tex]\[
\boxed{-0.5898}
\][/tex]
In summary, we have:
(a) The correlation coefficient between Happiness Index and the "Happy with Life" statement is [tex]\( \boxed{0.2493} \)[/tex].
(b) The correlation coefficient between Happiness Index and the "Won't Benefit" statement is [tex]\( \boxed{-0.5898} \)[/tex].
