ࡱ>  v|f {+bjbj 2#Htt8|tpcccc:pfcccc&%c Քd}09LpLa(t }: Section 4.1: Scatter Diagrams and Correlation Objectives: Students will be able to: Draw and interpret scatter diagrams Understand the properties of the linear correlation coefficient Compute and interpret the linear correlation coefficient Vocabulary: Response Variable variable whose value can be explained by the value of the explanatory or predictor variable Predictor Variable independent variable; explains the response variable variability Lurking Variable variable that may affect the response variable, but is excluded from the analysis Positively Associated if predictor variable goes up, then the response variable goes up (or vice versa) Negatively Associated if predictor variable goes up, then the response variable goes down (or vice versa) Key Concepts: Scatter Diagram Shows relationship between two quantitative variables measured on the same individual. Each individual in the data set is represented by a point in the scatter diagram. Explanatory variable plotted on horizontal axis and the response variable plotted on vertical axis. Do not connect the points when drawing a scatter diagram.  EMBED PowerPoint.Show.8  Properties of the Linear Correlation Coefficient The linear correlation coefficient is always between -1 and 1 If r = 1, then there is perfect positive linear relation between the two variables If r = -1, then there is perfect negative linear relation between the two variables The closer r is to 1, then the stronger the evidence for a positive linear relation The closer r is to -1, then the stronger the evidence for a negative linear relation If r is close to zero, then there is little evidence of a linear relation between the two variables. R close to zero does not mean that there is no relation between the two variables The linear correlation coefficient is a unitless measure of association Observational Data: If bivariate data are observational, then we cannot conclude that any relation between the explanatory and response variable are due to cause and effect  EMBED PowerPoint.Show.8  Example 1: 1 2 3 4 5 6 7 8 9 10 11 12 x 3 2 2 4 5 15 22 13 6 5 4 1 y 0 1 2 1 2 9 16 5 3 3 1 0  Draw a scatter plot of the above data  Compute the correlation coefficient Homework: pg 203 211; 4, 5, 11-16, 27, 38, 42 Section 4.2: Least-Squares Regression Objectives: Students will be able to: Find the least-squares regression line and us the line to make predictions Interpret the slope and the y-intercept of the least-squares regression line Compute the sum of squared residuals Vocabulary: Residual aka, the error; difference between observed value of y and the predicted value of y Method of Least-squares minimizes the sum of the residuals squared Least-squares regression line line that minimizes the sum of the squared errors Key Concepts:  EMBED PowerPoint.Show.8   EMBED PowerPoint.Show.8  Example : SEQ Table \* ARABIC 1 x 0123456789y 89.2 86.4 83.5 81.1 78.2 73.9 64.3 71.8 65.6 66.2 Find the least-squares regression line What does the model predict for x = 5? What is the residual from the above? Is this model appropriate for x = 15? Why or why not? Homework: pg 221 225; 2, 3, 6, 9, 19, 21 Section 4.3: Diagnostics on the Least-Squares Regression Line Objectives: Students will be able to: Compute and interpret the coefficient of determination Perform residual analysis on a regression model Identify influential observations Vocabulary: Coefficient of determination, R2 measures the percentage of total variation in the response variable that is explained by the least-squares regression line. Deviations differences between predicted value and actual value Total Deviation deviation between observed value, y, and mean of y, y-bar Explained Deviation deviation between predicted value, y-hat, and mean of y, y-bar Unexplained Deviation deviation between observed value, y, and predicted value, y-hat Influential Observation observation that significantly affects the value of the slope Key Concepts:  EMBED PowerPoint.Show.8  Is the Linear Model Appropriate? Patterned Residuals If a plot of the residuals against the explanatory variable shows a discernible pattern, such as a curve, then the response and explanatory variable may not be linearly related Variance of the Residuals Constant If a plot of the residuals against the explanatory variable shows the spread of the residuals increasing or decreasing as the explanatory variable increases, then a strict requirement of the linear model is violated. This requirement is called constant error variance. Influential Observations Influential observations typically exist when the point is an outlier relative to its X-value Outliers and Influential Observations Remove only if there is justification to do so Homework: pg 235 239; 2, 5, 9, 11-15, 29 Section 4.4: Nonlinear Regression: Transformations Objectives: Students will be able to: Change exponential expressions to logarithmic expressions and logarithmic expressions to exponential expressions Simplify expressions containing logarithms Use logarithmic transformations to linearize exponential relations Use logarithmic transformations to linearize power relations Vocabulary: Coefficient of determination, R2 measures the percentage of total variation in the response variable that is explained by the least-squares regression line. Deviations differences between predicted value and actual value Total Deviation deviation between observed value, y, and mean of y, y-bar Explained Deviation deviation between predicted value, y-hat, and mean of y, y-bar Unexplained Deviation deviation between observed value, y, and predicted value, y-hat Influential Observation observation that significantly affects the value of the slope Key Concepts: Some relationships that are nonlinear can be modeled with exponential or power models y = a bx (with b > 1) y = a bx (with b < 1) y = a xb  Exponential Model analyzed with Least Squares (Linear Regression) We started with an exponential model y = a bx We transformed that into a linear model Y = A + B X After we solve the linear model, we match up b = 10B a = 10A In this way, we are able to use the method of least-squares to find an exponential model Power Model analyzed with Least Squares (Linear Regression) We started with a power model y = a bx We transformed that into a linear model Y = A + B X After we solve the linear model, we find that b = B a = 10A In this way, we are able to use the method of least-squares to find a power model Homework: TBD Chapter 4: Review Objectives: Students will be able to: Summarize the chapter Define the vocabulary used Complete all objectives Successfully answer any of the review exercises Use the technology to compute required objectives Vocabulary: None new Problem 1: The scatter diagram to the right shows a Moderate positive linear relationship Weak negative linear relationship Strong positive linear relationship Strong nonlinear relationship  Problem 2: The scatter diagram to the left shows a Moderate positive linear relationship Weak negative linear relationship Strong positive linear relationship Strong nonlinear relationship Problem 3: The least squares line to the right could be Y = 1.5 X + 1 Y = X 2 Y = 0.5X 3 Y = 4X + 1 Problem 4: In a study of Y = weight in pounds versus X = age in months of certain dogs, the least squares regression line was found to be Y = 2.7 X + 1.7 The slope has the interpretation Newborn dogs, on the average, weigh 1.7 pounds Dogs, on the average, weigh 17 pounds at one year Newborn dogs, on the average, weight 2.7 pounds Dogs, on the average, gain 2.7 pounds per month Problem 5: A coefficient of determination R2 measures The slope of the least squares regression line The percent of total variation explained by the least squares regression line The relationship between the slope and the intercept of the least squares regression line The size of the residuals of the least squares regression line  Problem 6: The residual plot to the right shows that A linear model is inappropriate because of patterns The correlation is positive The intercept is negative A linear model is inappropriate because of slopes Problem 7: If a linear model is inappropriate because of a pattern in the residuals, one option to try to improve the model is to Calculate the coefficient of determination Switch the roles of X and Y Transform the X or the Y variable Ignore the residual plot Problem 8: Using logarithmic transforms to Y and/or to X, we are able to fit which types of models? 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