508 CHAPTER 10 Correlation and Regression 10-5 Nonlinear Regression • Use paired data to identify the linear, quadratic, logarithmic, exponential, and power models. • Determine which model best fits the paired data. Key Concept In Part 1 we introduce the linear correlation coefficient r, which is a number that measures how well paired sample data fit a straight-line pattern when graphed. We use the sample of paired data (sometimes called bivariate data) to find the value of r (usually using technology), and then we use that value to decide whether there is a linear correlation between the two variables. In this section we consider only linear relationships, which means that when graphed in a scatterplot, the points approximate a straight-line pattern. In Part 2, we discuss methods for conducting a formal hypothesis test that can be used to decide whether there is a linear correlation between all population values for the two variables. Finally, in Part 3 we discuss a method of randomization whereby we resample many times to test the null hypothesis of no correlation. 10-1 Correlation PART 1 Basic Concepts of Correlation We begin with the basic definition of correlation, a term commonly used in the context of an association between two variables. DEFINITIONS A correlation exists between two variables when the values of one variable are somehow associated with the values of the other variable. A linear correlation exists between two variables when there is a correlation and the plotted points of paired data result in a pattern that can be approximated by a straight line. Table 10-1, for example, includes paired sample data consisting of lottery jackpot amounts and numbers of tickets sold for nine different Powerball lotteries. We will determine whether there is a linear correlation between the variable x (jackpot amount) and the variable y (number of tickets sold). Instead of blindly jumping into the calculation of the linear correlation coefficient r, it is wise to first explore the data. Explore! Because it is always wise to explore sample data before applying a formal statistical procedure, we should use a scatterplot to graph the paired data in Table 10-1 and observe if there is a distinct pattern in the plotted points. (Scatterplots were first introduced in Section 2-4.) The scatterplot is shown in Figure 10-1 and there does appear to be a distinct pattern of increasing Powerball ticket sales corresponding to increasing jackpot amounts. There do not appear to be any outliers, which are data points that are far away from the other data points.
RkJQdWJsaXNoZXIy NjM5ODQ=