Notes 4.3: LSRL



Learning Target Calculate the Least Squares Regression Lines for Scatterplot Data LSRL: Least Squares Regression Line, also called the line of best fit. The line of best fit ALWAYS passes through the point ( x? , y? ). y-hat: (y?) Represents the predicted y value based on the LSRL. It is not an actual data point, but predicted based on the equation.y? = slope x + y-intercepty? = mx + b (in algebra)y? = a + bx (in statistics)Example 1: You can find ( x? , y? ) and then use a straight edge to try to come up with the LSRL that best fits every data point. We will want to know how to calculate the LSRL.( x? , y? ) 40005190500= (1955, 17583)Slope: A number that describes both the direction and steepness of a line.You know the formula for slope (from algebra) = riserun and you can estimate the slope of a LSRL by using 2 points. However in statistics, we will use a new formula to find the slope (based on ALL data points). Slope = correlation (standard deviation of ystandard deviation of x) = r SySx ?I can calculate the slope of a line of best fit using correlation ?Example 2: Use the formula to find the slope for a LSRL based on the data for fat and calories from the previous example.We have already computed Sx, Sy, and r, so the slope = (0.866) (64.255.38) = 10.34Interpretation: this indicates that as fat increases by 1 gram, calories increase by 10.34.Y-intercept: The value of y where the LSRL crosses the vertical axis (when x = 0).In algebra, you would find the y-intercept by using the equation for the line and solving for the y-intercept. It is also estimated by looking at the graph. ?I can calculate the y-intercept of a line of best fit ?To find the y-intercept that is based on ALL data points (for the LSRL), we will use another new formula: y-intercept = y? - slope (x?)Example 3: Use the formula to find the y-intercept for the fat and calorie data:y intercept = 167.5 – 10.34 ( 9.17) = 72.68Interpretation: When x = 0 (no fat grams), the calorie content is 72.68.Once you have calculated slope and y-intercept, you can write the equation for the LSRL:?I can determine the equation for a line of best fit (LSRL)?center0LSRL = y? = y-intercept + slope x00LSRL = y? = y-intercept + slope xThis equation should be interpreted in context (using words instead of variables).Example 4: Use the calculations that we have already done for the fat and calorie data to find the LSRLy? = 72.68 + 10.34 xor the predicted calorie content = 72.7 + 10.3 (fat grams)This is the LSRL for ALL the data points. Using y-intercept, slope, and ( x? , y?) you can graph the calculated LSRL on the scatterplot of fat and calories.Note: You could have all, a few, or no data points on the LSRL. ................
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