101 Ch. 8 and 9

101 Ch. 8 and 9 - Chapter 8 & 9 Linear Regression &...

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Linear Regression & Regression Wisdom
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Price of Homes Based on Size (in Square Feet) 600 800 1000 1200 1400 1600 1800 2000 2200 Price 1000 1500 2000 2500 3000 SQFT
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Models for Data Draw a line to summarize the relationship between two variables This line is called the regression line. Explanatory variable (x) Response variable (y)
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Correlation and the Line Price of Homes Based on Square Feet Price = -75.47 + 0.69SQFT R 2 = 80.2% 600 800 1000 1200 1400 1600 1800 2000 2200 Price 1000 1500 2000 2500 3000 SQFT
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Regression line Explains how the response variable ( y ) changes in relation to the explanatory variable ( x ) Use the line to predict value of y for a given value of x
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Regression line Need a mathematical formula We want to predict y from x The predicted values are called . The observed values are called y . y ˆ
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Regression line “Putting a hat on it” means we have predicted something from the model Look at vertical distance Amount of error in the regression line Place the regression line so these errors are as small as possible y y ˆ -
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Least squares regression Most commonly used regression line Makes the sum of the squared errors as small as possible Based on the statistics r s s y x y x , , , ,
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Regression line equation where x b b y 1 0 ˆ + = x y s s r b = 1 x b y b 1 0 - =
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Regression line equation b 1 = slope of line. For every unit increase in x , y changes by the amount of the slope. • Interpreting b 1 (slope): For every one unit increase in the explanatory variable , there will be a b 1 unit(s) increase/decrease in the response variable . For example: For every one square foot increase in size, there will be a $69 increase in home price. MEMORIZE THIS!!!!!
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Regression line equation b 0 = y -intercept of line. The value of y when x = 0. • Interpreting b 0 ( y -intercept): When the explanatory variable = 0 , the value of the response variable = b 0 . For example: When the sq. ft. is a home is 0, the price of the home will be -$7,547. MEMORIZE THIS!!!!! BE CAREFUL. The interpretation of the intercept does not always make sense. When interpreting, be sure to mention if the interpretation does not make sense.
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Example – Arby’s Sandwiches I visited Arby’s website and checked out the nutrition information that they had posted. I looked at the serving weight in grams and number of calories for 22 of their sandwiches. Let’s come up with the regression equation for this data.
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Calculating the regression line Some Useful Info: 7757 . 0 , 2 . 189 , 77 . 78 , 45 . 515 , 05 . 254 = = = = = r s s y x y x
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Calculating the Regression Line Find the Slope Find the Intercept 86 . 1 77 . 78 2 . 189 ) 7757 . 0 ( 1 = = = x y s s r b 92 . 42 ) 05 . 254 ( 86 . 1 45 . 515 1 0 = - = - = x b y b
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Calculating the regression line. Don’t forget to write the equation.
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This note was uploaded on 03/30/2008 for the course STAT 101 taught by Professor Graham during the Spring '08 term at Iowa State.

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101 Ch. 8 and 9 - Chapter 8 & 9 Linear Regression &...

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