Certainly! Let's create a scenario and interpret the line of best fit, [tex]\( y = 0.26x + 67.6 \)[/tex].
Scenario:
Imagine you are the owner of a small online business that tracks its monthly revenue in dollars. You have been collecting data on your monthly revenue over time to analyze how your business is growing. After plotting your data points and performing a linear regression analysis, you find that the line of best fit for your revenue data is given by the equation [tex]\( y = 0.26x + 67.6 \)[/tex], where [tex]\( y \)[/tex] represents your monthly revenue in dollars and [tex]\( x \)[/tex] represents the number of months since you started tracking your revenue.
Interpretation of the Line of Best Fit:
1. Slope (0.26):
The slope of the line is [tex]\( 0.26 \)[/tex]. This means that for each additional month ([tex]\( x \)[/tex]), your revenue ([tex]\( y \)[/tex]) increases by [tex]$0.26. In other words, the slope indicates the rate of growth of your revenue. Since the slope is positive, it suggests that your business is steadily growing. Specifically, each month your revenue increases by 26 cents.
2. Y-Intercept (67.6):
The y-intercept of the line is \( 67.6 \). This represents the starting value of your revenue when \( x = 0 \), which is the point in time when you began tracking your revenue. In this context, the y-intercept indicates that at the very beginning (the first month), your revenue was $[/tex]67.60. This value serves as the baseline revenue from which any changes (increases) are measured.
Practical Example:
- Month 0: At the start (month 0), your revenue is $67.60.
- Month 1: After one month, your revenue would be [tex]\( 0.26(1) + 67.6 = 67.86 \)[/tex] dollars.
- Month 6: After six months, your revenue would be [tex]\( 0.26(6) + 67.6 = 69.16 \)[/tex] dollars.
- Month 12: After one year, your revenue would be [tex]\( 0.26(12) + 67.6 = 70.72 \)[/tex] dollars.
This line of best fit allows you to forecast your future revenue growth based on the current trend and analyze the performance of your business over time.
