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Section 10.4

Section 10.4. Variation and Prediction Intervals. Introducing … . TOPIC ONE. Deviation vs. Variation. Can you measure the deviation and the variation for a pair of (x, y) values? If so, what’s the difference? . Not sure? Let’s Clarify!.

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Section 10.4

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  1. Section 10.4 Variation and Prediction Intervals

  2. Introducing … • TOPIC ONE

  3. Deviation vs. Variation • Can you measure the deviation and the variation for a pair of (x, y) values? • If so, what’s the difference?

  4. Not sure? Let’s Clarify! • Total Deviation: y – y (given value for y minus the average value for y) • Explained Deviation: y – y (predicted value for y minus the average value for y). • Unexplained Deviation: y – y (given value for y minus the predicted value for y).

  5. Putting Words to Images • Total Deviation: y – y (given value for y minus the average value for y) • Explained Deviation: y – y (predicted value for y minus the average value for y). • Unexplained Deviation: y – y (given value for y minus the predicted value for y).

  6. Let’s Practice • You are given the following: • The equation of the regression line is y = 3 + 2x • The mean of the y-values is 9. • One of the pairs of sample data is (5, 19). • Find the total deviation, explained deviation, and unexplained deviation.

  7. The connection • The total variation is the sum of the squares of the total deviation values. • The explained variation is the sum of the squares of the explained deviation values. • The unexplained variation is the sum of the squares of the unexplained deviation values. • If we sum the squares of deviation values we get amounts of variation.

  8. The connection (Total variation) = (explained variation) + (unexplained variation)

  9. Let’s Practice • Find the explained variation, unexplained variation, and total variation for the following data set: Listed above are the overhead widths (in cm) of seals measured from photographs and the weights of the seals (in kg).

  10. One Step Further • The coefficient of determination is the amount of the variation in y that is explained by the regression line. r² = Explained variation Total variation

  11. How do we use this now? • In Section 10-2 we used paired subway and pizza costs in NY to determine the correlation coefficient r = 0.988. • Find the coefficient of determination and then use this to find the percentage of total variation that can be explained by the linear relationship between the cost of a slice of pizza and the cost of subway fare.

  12. How do we use this now? • Use the value of linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables. • r = -0.865 (x = car weight, y = city fuel consumption in mi/gal)

  13. How do we use this now? • Use the value of linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables. • r = -0.488 (x = age of home, y = home selling price)

  14. Connecting the Dots • What are two things that you think or wish there was a correlation between? • Select two things for their to be a correlation between (this is your choice – it can be made-up). • Select an r and describe what the correlation in your own words (this is your choice – it can be made-up). • Now based on your example, find the coefficient of determination and describe the relationship that exisits.

  15. For Example … • I think there might be a correlation between the number of minutes one spends commuting to work and stress level. • If I did the necessary calculations and found that there was a correlation and r = -0.678, then this would mean …

  16. YOUR TURN! • What are two things that you wish there was a correlation between? • Select two things for their to be a correlation between (this is your choice – it can be made-up). • Select an r and describe what the correlation in your own words (this is your choice – it can be made-up). • Now based on your example, find the coefficient of determination and describe the relationship that exisits.

  17. Putting It All Together • Find the explained variation, unexplained variation, total variation, and coefficient of determination for the following data set. Listed above are concentrations (in parts per million) of CO2 and temperatures (in ◦C) for different years.

  18. Homework • Pg. 557-559 #4-6, 13 (skip part e), 14 (skip part e)

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