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Statistics Class 8 2/13/2013 Quiz 18 Listed below are brain sizes (in 3 ) and Wechsler IQ scores of subjects. Is there sufficient evidence to conclude that there is a linear correlation between brain size and IQ Score? Does it appear that people with larger brains are more intelligent? Brain 965 1029 1030 1285 1049 1077 1037 1068 1176 1105 IQ 85 86 102 103 97 124 125 102 114 90 Regression Given a collection of sample data, the regression equation = + 1 Algebraically describes the relationship between the two variables x and y. The graph of the regression equation is called the regression line (or line of best fit, or least-squares line). x is called the predictor variable, or Independent variable. is called the response variable, or dependent variable Regression Given a collection of sample data, the regression equation = + 1 Algebraically describes the relationship between the two variables x and y. The graph of the regression equation is called the regression line (or line of best fit, or least-squares line). x is called the explanatory variable, or independent variable. is called the response variable, or dependent variable The regression equation should remind you of = + Regression Requirements 1. The sample of paired (x,y) data is a random sample of quantitative data. 2. Visual examination of the scatterplot shows that the points approximate a straight line pattern. 3. Outliers can have a strong effect on the regression, so remove any outliers that are known to be errors. Consider the effects of any outliers that are not known errors Regression Formulas • 1 = −( )( ) 2 −( )2 • 0 = ( ) 2 −( )( ) 2 −( )2 • 1 = , where is the standard deviation of the y values and is the standard deviation of the x’s • 0 = − 1 Regression Formulas • 1 = −( )( ) 2 −( )2 • 0 = ( ) 2 −( )( ) 2 −( )2 • 1 = , where is the standard deviation of the y values and is the standard deviation of the x’s • 0 = − 1 Rounding Round to 3 significant digits. Regression Cost of Pizza 0.15 0.35 1.00 1.25 1.75 2.00 Subway Fare 0.15 0.35 1.00 1.35 1.50 2.00 Regression Cost of Pizza 0.15 0.35 1.00 1.25 1.75 2.00 Subway Fare 0.15 0.35 1.00 1.35 1.50 2.00 We are using the Pizza/Subway data from last time. Where pizza is the x variable (explanatory) and the subway fare is the y variable (response). 1. Put the data into the Calculator. Regression Cost of Pizza 0.15 0.35 1.00 1.25 1.75 2.00 Subway Fare 0.15 0.35 1.00 1.35 1.50 2.00 We are using the Pizza/Subway data from last time. Where pizza is the x variable (explanatory) and the subway fare is the y variable (response). 1. Put the data into the Calculator. 2. Hit the [STAT] button, then [→], then [4] or select LinReg(ax+b). Regression Cost of Pizza 0.15 0.35 1.00 1.25 1.75 2.00 Subway Fare 0.15 0.35 1.00 1.35 1.50 2.00 We are using the Pizza/Subway data from last time. Where pizza is the x variable (explanatory) and the subway fare is the y variable (response). 1. Put the data into the Calculator. 2. Hit the [STAT] button, then [→], then [4] or select LinReg(ax+b). 3. Skip down to Calculate and hit [ENTER]. For us a is and b is . Regression Cost of Pizza 0.15 0.35 1.00 1.25 1.75 2.00 Subway Fare 0.15 0.35 1.00 1.35 1.50 2.00 1. Put the data into the Calculator. 2. Hit the [STAT] button, then [→], then [4] or select LinReg(ax+b). 3. Skip down to Calculate and hit [ENTER]. For us a is and b is . 4. Make a scatter plot. Regression Cost of Pizza 0.15 0.35 1.00 1.25 1.75 2.00 Subway Fare 0.15 0.35 1.00 1.35 1.50 2.00 1. Put the data into the Calculator. 2. Hit the [STAT] button, then [→], then [4] or select LinReg(ax+b). 3. Skip down to Calculate and hit [ENTER]. For us a is and b is . 4. Make a scatter plot. 5. Press [Y=] with the flashing box in the the 1 spot, hit the [VARS] button then [5] then [→] twice so that you are in the EQ menu. Press [1] or select REGEQ. 6. Press [GRAPH]. Regression Using the Regression Equation for Predictions 1. Use the regression equation only if the graph of the regression line on the scatterplot fits the data reasonably well. Regression Using the Regression Equation for Predictions 1. Use the regression equation only if the graph of the regression line on the scatterplot fits the data reasonably well. 2. Use the regression equation only if there is a linear correlation. Regression Using the Regression Equation for Predictions 1. Use the regression equation only if the graph of the regression line on the scatterplot fits the data reasonably well. 2. Use the regression equation only if there is a linear correlation. 3. Use the regression equation for predictions that do not go very far past the sample data. Regression Using the Regression Equation for Predictions 1. Use the regression equation only if the graph of the regression line on the scatterplot fits the data reasonably well. 2. Use the regression equation only if there is a linear correlation. 3. Use the regression equation for predictions that do not go very far past the sample data. 4. If the regression equation does not appear to be useful for making predictions the best predicted value of a variable is its point estimate, which is its sample mean. Regression Using the Regression Equation for Predictions 1. Use the regression equation only if the graph of the regression line on the scatterplot fits the data reasonably well. 2. Use the regression equation only if there is a linear correlation. 3. Use the regression equation for predictions that do not go very far past the sample data. 4. If the regression equation does not appear to be useful for making predictions the best predicted value of a variable is its point estimate, which is its sample mean. Try! The paired value of the Consumer Price Index (CPI) and the cost of subway fare are listed below. Find the best predicted cost of Subway fare when the Consumer Price index is 182.5 CPI 30.2 48.3 112.3 162.2 191.9 197.8 Subway Fare 0.15 0.35 1.00 1.35 1.50 2.00 Try Some more! Find the best predicted temperature (in °F) at a time when cricket chirps 3000 times in one minute. What is wrong with this predicted value? Chirps 882 1188 1104 864 1200 1032 960 900 Temp (°F) 69.7 93.3 76.3 88.6 71.6 79.6 84.3 82.6 HOMEWORK!! • 10-3: 1-9 odd, 13-17 odd, 23, 25, 27.