Sure! Let's analyze the data provided and find the required information step-by-step.
### Step 1: Define the Variables
Given:
- Years: 2007, 2008, 2009, 2011, 2013
- Attendance (in millions): 8.3, 8.5, 8.5, 8.8, 9.3
Let [tex]\( x \)[/tex] represent the years such that [tex]\( x=0 \)[/tex] corresponds to 2007.
So, the transformed years are:
- 2007 → [tex]\( x = 0 \)[/tex]
- 2008 → [tex]\( x = 1 \)[/tex]
- 2009 → [tex]\( x = 2 \)[/tex]
- 2011 → [tex]\( x = 4 \)[/tex]
- 2013 → [tex]\( x = 6 \)[/tex]
The attendance values are:
[tex]\[ y = [8.3, 8.5, 8.5, 8.8, 9.3] \][/tex]
### Step 2: Calculate the Linear Regression Equation
Based on the data points, we can determine the linear regression equation of the form:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\( m \)[/tex] is the slope
- [tex]\( b \)[/tex] is the y-intercept
For this case, the slope [tex]\( m \)[/tex] is calculated as:
[tex]\[ m \approx 0.16 \][/tex]
The y-intercept [tex]\( b \)[/tex] is calculated as:
[tex]\[ b \approx 8.27 \][/tex]
Thus, the linear regression equation, rounding to the nearest hundredth, is:
[tex]\[ y = 0.16x + 8.27 \][/tex]
### Step 3: Calculate the Correlation Coefficient
The correlation coefficient [tex]\( r \)[/tex] is a measure of the strength and direction of the linear relationship between the variables.
For this data set, the correlation coefficient [tex]\( r \)[/tex] is calculated as:
[tex]\[ r \approx 0.97 \][/tex]
### Step 4: Determine the Association Strength
Based on the correlation coefficient:
- An absolute value of [tex]\( r \)[/tex] closer to 1 indicates a stronger linear relationship.
- An absolute value of [tex]\( r \)[/tex] closer to 0 indicates a weaker linear relationship.
Since [tex]\( |0.97| \)[/tex] is very close to 1, we can conclude that there is a strong linear association between the years and attendance.
### Final Answer
The linear regression equation is:
[tex]\[ y = 0.16x + 8.27 \][/tex]
The correlation coefficient is:
[tex]\[ r = 0.97 \][/tex]
This suggests that the data have a strong association.
