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STRENGTH: CORRELATION = r xi x 1 Correlation r n 1 sx yi y sy Is there a relationship between student quiz grades and their test grade? Quiz Avg. 75 86 92 95 80 Test Avg. 79 86 100 95 90 Describe the association’s Form, Direction, and Strength 95 80 x 85.6 Test Avg. 79 86 100 95 90 y 90 Quiz Avg. 75 86 92 xi x 1 r n 1 sx r sx 8.26 s y 8.09 yi y sy 1 75 85.6 79 90 86 85.6 86 90 92 85.6 100 90 5 1 8.26 8.09 8.26 8.09 8.26 8.09 95 85.6 95 90 80 85.6 90 90 8.26 8.09 8.26 8.09 r = ¼ [1.7449 + -0.0239 + 0.9577 + 0.7033 + 0] r = ¼ (3.382) = 0.8455 WARM-UP Just put these data points into List 1,2,3,& 4 Describe the Form, Direction, and Strength of each. A. X Y 0 B. X Y 0 0 50 1 5 1 48 2 11 2 30 3 20 3 40 4 32 4 20 5 58 5 35 6 120 6 10 7 200 7 12 r = 0.88 r = -0.87 Chapter 7 (continued) FACTS ABOUT CORRELATION 1. Positive r refers to positively associated variables while negative r refers to negatively associated variables. (The Pos./Neg. sign of ‘r’ matches the slope’s sign.) 2. Correlation is ALWAYS between -1 ≤ r ≤ 1. The correlation is strong when r is close to 1 or -1 but weak when r is close to zero. 3. r has NO UNITS. 4. Correlation is only valuable for LINEAR relationships. 5. Like the Mean and Std. Dev., Correlation is non-resistant and is very influenced by outliers. CHAPTER 8 - Interpreting the Least Squares Regression Model is: ŷ = a + bx The The ŷ b is called y-hat and represents the Predicted y values. is the Slope of the linear equation: Interpreted as: For each unit increase in x the y-variable is predicted to change b amount on average. The a is the y-intercept of the linear equation: Interpreted as: ”a” is the average amount of the y variable when x = 0. IceCream Sales = 35 + 12.50(Temperature) R-Squared – the percent or fraction of variation in the values of y that is explained by the least squares regression of y on x. R-Squared – Is also called the Coefficient of Determination. It identifies what percent of the variation in the predicted values of y that are attributed by x. Thus (100% - R2) of the variation in y is attributed to by other factors. IceCream Sales = 35 + 12.50(Temperature) Does temperature outside affect the number of ice cream treats a store sells. R2 = 88.5% R2: 88.5% of the variation is the predicted amount of ice cream sales is attributed by the temperature outside. Fat (g) 19 Calories 920 31 34 35 1300 1310 960 39 1180 26 43 1100 1260 Regression model: Calories = 785.94 + 11.14(Fat) Slope: For every additional gram of fat the model predicts approximately an additional 11.14 Calories in the food. y-intercept: If the food product contained NO fat it would still have 785.94 calories on average. Correlation: r = 0.562 Moderate strong positive linear relationship. Residual plot: plot R2: 31.5% of the variation in the predicted amount of calories is attributed by the amount of fat. Homework: Page 189: 7, 8, 9, 10, [13, 14 omit c] Fast food is often considered unhealthy because of the amount of fat and calories in it. Does the amount of Fat content contribute to the number of calories a food product contains? Fat (g) 19 Calories 920 31 34 35 1300 1310 960 39 1180 26 43 1100 1260 1. Construct and Describe the Scatterplot for this data. 2. Find the Regression model and interpret the slope, yintercept, correlation, residual plot, and R2.