University:
High School
Course:
High School Mathematics
Academic year:

2022
Views:

317

Pages:

2
Author:

Amelie Villegas

Analyzing Lines of Fit Core Concepts Residuals A residual is the difference of the y-value of a data point and the corresponding y-value found using the line of fit. A residual can be positive, negative, or zero. How far the point is from the equation A scatter plot of the residuals shows how well a model fits a data set. If the model is a good fit, then the absolute values of the residuals are relatively small, and the residual points will be more or less evenly dispersed about the horizontal axis. If the model is not a good fit, then the residual points will form some type of pattern that suggests the data are not linear. Wildly scattered residual points suggest that the data might have no correlation. linear regression (residuals) a feature on the calculator that figures out the line of fit line of best fit an equation that will explain the data relationship correlation coefficient = r explains direction How tight the points are -1 ≤ r ≤ +1 Strong Negative Strong Positive Weak Fit +1 +.95 -.75 -.15 Problem 1: Equation: y = -3x + 2 Data Table: | x | y | |---|---| | -2 | 13 | | -1 | 11 | | 0 | 8 | | 1 | 5 | | 2 | 2 | | 3 | -1 | | 4 | -4 | | 5 | -7 | | 6 | -10 | Residuals: | x | y | Predicted y | Residual (Difference) | |---|---|---|---| | -2 | 13 | 14 | -1 | | -1 | 11 | 11 | 0 | | 0 | 8 | 8 | 0 | | 1 | 5 | 5 | 0 | | 2 | 2 | 2 | 0 | | 3 | -1 | -1 | 0 | | 4 | -4 | -4 | 0 | | 5 | -7 | -7 | 0 | | 6 | -10 | -10 | 0 | Analysis: Adding up all the residuals, we have -1. This is close to 0 so the equation is a good fit for the data. Problem 2: Equation: y = -0.5x + 1 Data Table: | x | y | |---|---| | 0 | 2 | | 1 | 1.5 | | 2 | 1 | | 3 | 0 | | 4 | -1 | | 5 | -2 | | 6 | -3 | | 7 | -4 | | 8 | -5 | Residuals: | x | y | Predicted y | Residual (Difference) | |---|---|---|---| | 0 | 2 | 1 | 1 | | 1 | 1.5 | 0.5 | 1 | | 2 | 1 | 0 | 1 | | 3 | 0 | -0.5 | 0.5 | | 4 | -1 | -1 | 0 | | 5 | -2 | -1.5 | -0.5 | | 6 | -3 | -2 | -1 | | 7 | -4 | -2.5 | -1.5 | | 8 | -5 | -3 | -2 | Analysis: Looking at all the residuals, under the axis, this equation is a bad fit for the data. Problem 3: Table: | Average Daily Temperature (°F) | Number of Beach Visitors | |---|---| | 80 | 100 | | 82 | 150 | | 83 | 145 | | 85 | 190 | | 86 | 215 | | 88 | 263 | | 89 | 300 | a. Plot the data and find the line of best fit. A scatter plot is shown with temperature on the x-axis and number of visitors on the y-axis. The line of best fit is drawn through the data points. Equations: y - 100 = 20.375(x - 80) y - 100 = 20.375x - 1630 y = 20.375x - 1530 b. Identify and interpret the correlation coefficient. Since the points are close to the line, I would say it's a strong positive correlation. c. Interpret the slope and y-intercept of the line of best fit. Slope: About 20 more people come to the beach for every 1 degree warmer the temperature is. Y-intercept: -1530 people will be at the beach if the temperature is 0 degrees.