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Answer :
To solve this problem, we need to find two key things: the correlation coefficient and the proportion of variation in [tex]\( y \)[/tex] explained by [tex]\( x \)[/tex], known as [tex]\( R^2 \)[/tex].
### Correlation Coefficient
1. Definition: The correlation coefficient, denoted as [tex]\( r \)[/tex], is a measure of the strength and direction of a linear relationship between two variables. The value of [tex]\( r \)[/tex] ranges from -1 to 1.
- If [tex]\( r = 1 \)[/tex], there is a perfect positive linear relationship.
- If [tex]\( r = -1 \)[/tex], there is a perfect negative linear relationship.
- If [tex]\( r = 0 \)[/tex], there is no linear relationship.
2. Calculation:
- To find [tex]\( r \)[/tex], we consider all the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values given in the data set.
- We use a statistical method (such as Pearson correlation) to calculate [tex]\( r \)[/tex]. This involves calculating the covariance of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], and the standard deviations of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
3. Result:
- The calculated correlation coefficient for this data set is [tex]\( r = -0.559 \)[/tex].
### Proportion of Variation Explained ([tex]\( R^2 \)[/tex])
1. Definition: [tex]\( R^2 \)[/tex], the coefficient of determination, represents the proportion of the variance in the dependent variable ([tex]\( y \)[/tex]) that is predictable from the independent variable ([tex]\( x \)[/tex]).
2. Calculation:
- [tex]\( R^2 \)[/tex] is simply the square of the correlation coefficient: [tex]\( R^2 = r^2 \)[/tex].
- We take the computed correlation coefficient and square it, then convert it into a percentage to understand the explained variance in terms of percentage.
3. Result:
- Squaring our correlation coefficient [tex]\( -0.559 \)[/tex], we get [tex]\( R^2 = (-0.559)^2 = 0.312 \)[/tex].
- As a percentage, this is [tex]\( R^2 = 31.2\% \)[/tex].
Therefore, the correlation coefficient [tex]\( r \)[/tex] is [tex]\(-0.559\)[/tex], and [tex]\( 31.2\%\)[/tex] of the variation in [tex]\( y \)[/tex] can be explained by the variation in [tex]\( x \)[/tex].
### Correlation Coefficient
1. Definition: The correlation coefficient, denoted as [tex]\( r \)[/tex], is a measure of the strength and direction of a linear relationship between two variables. The value of [tex]\( r \)[/tex] ranges from -1 to 1.
- If [tex]\( r = 1 \)[/tex], there is a perfect positive linear relationship.
- If [tex]\( r = -1 \)[/tex], there is a perfect negative linear relationship.
- If [tex]\( r = 0 \)[/tex], there is no linear relationship.
2. Calculation:
- To find [tex]\( r \)[/tex], we consider all the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values given in the data set.
- We use a statistical method (such as Pearson correlation) to calculate [tex]\( r \)[/tex]. This involves calculating the covariance of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], and the standard deviations of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
3. Result:
- The calculated correlation coefficient for this data set is [tex]\( r = -0.559 \)[/tex].
### Proportion of Variation Explained ([tex]\( R^2 \)[/tex])
1. Definition: [tex]\( R^2 \)[/tex], the coefficient of determination, represents the proportion of the variance in the dependent variable ([tex]\( y \)[/tex]) that is predictable from the independent variable ([tex]\( x \)[/tex]).
2. Calculation:
- [tex]\( R^2 \)[/tex] is simply the square of the correlation coefficient: [tex]\( R^2 = r^2 \)[/tex].
- We take the computed correlation coefficient and square it, then convert it into a percentage to understand the explained variance in terms of percentage.
3. Result:
- Squaring our correlation coefficient [tex]\( -0.559 \)[/tex], we get [tex]\( R^2 = (-0.559)^2 = 0.312 \)[/tex].
- As a percentage, this is [tex]\( R^2 = 31.2\% \)[/tex].
Therefore, the correlation coefficient [tex]\( r \)[/tex] is [tex]\(-0.559\)[/tex], and [tex]\( 31.2\%\)[/tex] of the variation in [tex]\( y \)[/tex] can be explained by the variation in [tex]\( x \)[/tex].
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