Final answer:
A regression equation is used to predict the relationship between an independent variable (x) and a dependent variable (y). The line of best fit is determined using least-squares regression and expressed as ý = a + bx, with 'a' as the y-intercept and 'b' as the slope. The correlation coefficient indicates the strength of the linear relationship.
Step-by-step explanation:
To write the regression equation, we first need to identify the independent and dependent variables. In our example, the independent variable (x) is the pinky finger length, as it is the variable we are using to make predictions, and the dependent variable (y) is the person's height, as it is the variable we are trying to predict. Start by plotting these variables on a scatter plot to visualize any potential linear relationships.
After plotting the data, if we see a trend that suggests a linear relationship, we can use regression analysis to find the line of best fit. The least-squares regression line minimizes the sum of the squares of the residuals, providing the most accurate linear model for our data. The line of best fit is usually expressed in the form ý = a + bx, where 'a' is the y-intercept and 'b' is the slope of the line.
To find the precise equation, we can use technology, such as a graphing calculator or statistical software, which can calculate the line of best fit by inputting the data. Then we sketch this line onto the same axes as our scatter plot to visually assess the fit. Additionally, we can calculate the correlation coefficient (r), which measures the strength and direction of the linear relationship between x and y. A correlation coefficient close to 1 or -1 signifies a strong linear relationship, whereas a value close to 0 indicates a weak relationship.
The predicted height for a pinky length of 2.5 inches can be estimated by substituting x=2.5 into our regression equation. Finally, it is important to remember that while regression lines can be used to predict values within the given set of data, they should not be extrapolated to predict values outside the set.