Final answer:
The question involves deriving the equation for the line of best fit from a data set of average annual temperatures at different elevations. This involves drawing a scatter plot, calculating the best-fit line using least-squares regression, and evaluating the significance of the correlation coefficient. Once calculated, the line allows predictions of temperatures at various elevations.
Step-by-step explanation:
The student's question involves analyzing a set of data points representing average annual temperatures collected at different elevations in the United States and deriving the equation for the line of best fit. To tackle this task, we start by determining which variable will serve as the independent variable (elevation) and which will be the dependent variable (temperature). A scatter plot is drawn to visualize the relationship between these two variables.
Through visual inspection of the scatter plot, a line of best fit is sketched—this does not have to touch every data point but should follow the general trend of the data. To quantitatively determine the line of best fit, we can calculate the least-squares regression line using a statistical calculator. The resulting equation typically takes the form ý = a + bx, where 'a' represents the y-intercept and 'b' is the slope of the line.
The correlation coefficient is also calculated to assess the strength and direction of the linear relationship between elevation and temperature. If the correlation coefficient is significant, it suggests a strong linear relationship. Finally, with the equation of the line of best fit, predictions can be made for the average annual temperatures at given elevations, exemplifying the importance of statistical analysis in understanding and interpreting real-world data.