#### Transcript Get out p. 193 HW and notes

Get out p. 193 HW and notes LEAST-SQUARES REGRESSION 3.2 Interpreting Computer Regression Output Interpreting Computer Regression Output A number of statistical software packages produce similar regression output. Be sure you can locate • the slope b • the y intercept a • the values of s and r2 Interpreting Computer Regression Output A number of statistical software packages produce similar regression output. Be sure you can locate • the slope b • the y intercept a • the values of s and r2 Example, p. 181 & 182 A random sample of 15 high school students was selected from the U.S. CensusAtSchool database. The foot length (in cm) and height (in cm) of each student in the sample were recorded. Example, p. 181 & 182 (a) What is the equation of the least-squares regression line that describes the relationship between foot length and height? Define any variables that you use. 𝑦 = 103.41 + 2.7469𝑥 where x = foot length and y = height. OR ℎ𝑒𝑖𝑔ℎ𝑡 = 103.41 + 2.7469(𝑓𝑜𝑜𝑡 𝑙𝑒𝑛𝑔𝑡ℎ) Example, p. 181 & 182 (c) Find the correlation. Take the square root of r2 = .486. 𝑟 2 = .486 ≈ ±0.697 Because the scatterplot showed a positive relationship, r = 0.697. Regression to the Mean How to Calculate the Least-Squares Regression Line We have data on an explanatory variable x and a response variable y for n individuals. From the data, calculate the means and the standard deviations of the two variables and their correlation r. The least-squares regression line is the line ŷ = a + bx with slope sy b=r sx And y intercept a = y - bx Example, p. 183 Using Feet to Predict Height. The mean and standard deviations of the foot lengths are 𝑥 = 24.76 cm and 𝑠𝑥 = 2.71 cm. The mean and standard deviation of the heights are 𝑦 = 171.43 cm and 𝑠𝑦 = 10.69 cm. The correlation between foot length and height is 𝑟 = 0.697. Problem: Find the equation for the least-squares regression line for predicting height from foot length. Show your work. Slope: 𝑏 = 𝑟 𝑠𝑦 𝑠𝑥 = 0.697 10.69 2.71 ≈ 2.7494 Y-intercept: 𝑎 = 𝑦 − 𝑏𝑥 = 171.43 − 2.7494(24.76) ≈ 103.3549 LSRL: 𝑦 = 103.3549 + 2.7494𝑥 , where x = foot length and y = height. Correlation and Regression Wisdom 1. The distinction between explanatory and response variables is important in regression. Correlation and Regression Wisdom 2. Correlation and regression lines describe only linear relationships. r = 0.816. r = 0.816. r = 0.816. r = 0.816. Correlation and Regression Wisdom 3. Correlation and least-squares regression lines are not resistant. Correlation and Regression Wisdom 4. Association does not imply causation. Outliers and Influential Observations in Regression An outlier is an observation that lies outside the overall pattern of the other observations. Points that are outliers in the y direction but not the x direction of a scatterplot have large residuals. Other outliers may not have large residuals. An observation is influential for a statistical calculation if removing it would markedly change the result of the calculation. Points that are outliers in the x direction of a scatterplot are often influential for the least-squares regression line. Outliers Influential Example p. 190 • The strong influence of Child 18 makes the original regression of Gesell score on age at first word misleading. The original data have r2 = 0.41, which means the age a child begins to talk explains 41% of the variation on a later test of mental ability. This relationship is strong enough to be interesting to parents. If we leave out Child 18, r2 drops to only 11%. The apparent strength of the association was largely due to a single influential observation. HW Due: Friday • P. 196 #59, 61 a, 63, 72, 73