Sure, let's work through the problem step by step.
### a. Finding the Correlation Coefficient
The correlation coefficient (denoted as [tex]\( r \)[/tex]) measures the strength and direction of the linear relationship between two variables. The values of [tex]\( r \)[/tex] range between -1 and 1. When [tex]\( r \)[/tex] is close to 1 or -1, there is a strong linear relationship between the variables. When [tex]\( r \)[/tex] is 0, there is no linear relationship.
Given:
- Hours of mixing (X): 2, 4, 6, 8, 10, 12
- Temperature of wood pulp (Y): 21, 27, 29, 64, 86, 92
Based on the calculations, the correlation coefficient between the variables [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] is found to be:
[tex]\[ r = 0.9577506853493282 \][/tex]
Interpretation:
A correlation coefficient of approximately 0.958 indicates a very strong positive linear relationship between the hours of mixing and the temperature of the wood pulp. In other words, as the number of hours of mixing increases, the temperature of the wood pulp also tends to increase strongly and consistently.
### b. Finding the Line of Regression of [tex]\( y \)[/tex] on [tex]\( x \)[/tex]
To find the line of regression of [tex]\( y \)[/tex] on [tex]\( x \)[/tex], we need to calculate the slope ([tex]\( b \)[/tex]) and the intercept ([tex]\( a \)[/tex]) of the regression line. The equation of the regression line can be written as:
[tex]\[ Y = a + bX \][/tex]
Based on the calculations, we have found:
- Slope ([tex]\( b \)[/tex]): [tex]\( 8.1 \)[/tex]
- Intercept ([tex]\( a \)[/tex]): [tex]\( -3.5333333333333314 \)[/tex]
Therefore, the equation of the regression line is:
[tex]\[ Y = 8.1X - 3.5333333333333314 \][/tex]
### c. Predicting the Temperature of Wood Pulp when [tex]\( X = 18 \)[/tex]
To predict the temperature of wood pulp [tex]\( (Y) \)[/tex] when [tex]\( X = 18 \)[/tex], we simply substitute [tex]\( X = 18 \)[/tex] into the regression equation:
[tex]\[ Y = 8.1(18) - 3.5333333333333314 \][/tex]
By performing the calculation:
[tex]\[ Y = 145.8 - 3.5333333333333314 \][/tex]
[tex]\[ Y = 142.26666666666665 \][/tex]
Therefore, the predicted temperature of the wood pulp when X is 18 hours of mixing is approximately 142.27 degrees.
These results give us a comprehensive understanding of the relationship between the hours of mixing and the temperature of the wood pulp, as well as a method to predict the temperature based on the number of hours of mixing.
