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CHAPTER 12: LINEAR REGRESSION AND CORRELATION

CHAPTER 12: LINEAR REGRESSION AND CORRELATION. Chapter Objectives By the end of this chapter, the student should be able to: • Discuss basic ideas of linear regression and correlation. • Create and interpret a line of best fit. • Calculate and interpret the correlation coefficient.

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CHAPTER 12: LINEAR REGRESSION AND CORRELATION

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  1. CHAPTER 12: LINEAR REGRESSION AND CORRELATION Chapter Objectives By the end of this chapter, the student should be able to: • Discuss basic ideas of linear regression and correlation. • Create and interpret a line of best fit. • Calculate and interpret the correlation coefficient. • Calculate and interpret outliers.

  2. 12.1 – Linear Equations Linear regression for two variables is based on a linear equation with one independent variable. The equation has the form: The variable x is the and y is the . y = a + bx independent variable dependent variable Linear equations of this form occur in applications of life sciences, social sciences, psychology, business, economics, physical sciences, mathematics, and other areas.

  3. Example 1 Aaron's Word Processing Service (AWPS) does word processing. The rate for services is $32 per hour plus a $31.50 one-time charge. The total cost to a customer depends on the number of hours it takes to complete the job. Find the equation that expresses the total cost in terms of the number of hours required to complete the job. Let x = Let y = Example 2 Emma’s Extreme Sports hires hang-gliding instructors and pays them a fee of $50 per class as well as $20 per student in the class. The total cost Emma pays depends on the number of students in a class. Find the equation that expresses the total cost in terms of the number of students in a class. Number of hours on the job y = 31.50 + 32x Total cost to complete the job Number of students in the class Let x = Let y = y = 50 + 20x Total cost Emma pays instructors

  4. Slope and Y-Intercept of a Linear Equation For the linear equation y = a + bx, b = and a = . From algebra, recall that slope is a number that describes the steepness of a line, and the y-intercept is the y-coordinate of the point (0, a) where the line crosses the y-axis. There are three possible graphs of y = a + bx: 1) 2) 3) y-intercept slope

  5. Example 3 Svetlana tutors to make extra money for college. For each tutoring session, she charges a one-time fee of $25 plus $15 per hour of tutoring. A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is y = 25 + 15x. What are the independent and dependent variables? What is the y-intercept and what is the slope? Interpret them using complete sentences. Independent Variable- Dependent Variable- # of hours spent tutoring per session Total amount she earns per session The y-intercept is 25. This represents the amount she’d get paid after tutoring for 0 hours. It doesn’t make sense in the context of the situation. The slope is 15 which represents that for each additional hour of tutoring, Svetlana earns $15.

  6. Example 4 Ethan repairs household appliances like dishwashers and refrigerators. For each visit, he charges $25 plus $20 per hour of work. A linear equation that expresses the total amount of money Ethan earns per visit is y= 25 + 20x. What are the independent and dependent variables? What is the y-intercept and what is the slope? Interpret them using complete sentences. Independent Variable- Dependent Variable- # of hours spent repairing an appliance Total amount he earns per visit The y-intercept is 25. This represents the amount Ethan gets paid just for coming to your house. In other words, even if he can’t fix your appliance, you will be charged $25. The slope is 20 which represents that for each additional hour spent on repairs, Ethan earns $20.

  7. Example 5 A vacation resort rents SCUBA equipment to certified divers. The resort charges an up-front fee of $25 and another fee of $12.50 an hour. a) What are the dependent and independent variables? b) Find the equation that expresses the total fee in terms of the number of hours the equipment is rented. c) Interpret the slope and y-intercept in the context of the situation. Independent Variable- Dependent Variable- # of hours spent using the SCUBA equipment Total cost to the customer y = 25 + 12.50x The slope is 12.50 which represents that for each additional hour spent scuba diving, you will be charged $12.50. The y-intercept is 25. This represents the fee just to rent the equipment. Technically, it’s the cost if you scuba dive for 0 hours which doesn’t make sense.

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