534 CHAPTER 10 Correlation and Regression a. Use the jackpot>tickets data from Table 10-1 on page 507 to predict the number of lottery tickets sold when the jackpot is $625 million. How close is the predicted value to the actual value of 90 million tickets that were actually sold when the Powerball lottery had a jackpot of $625 million? b. Predict the IQ score of an adult who is exactly 175 cm tall. CP YOUR TURN. Do Exercise 5 “Cars.” EXAMPLE 4 Making Predictions SOLUTION a. Good Model: Use the Regression Equation for Predictions. The regression line fits the points well, as shown in Example 3. Also, there is a linear correlation between Powerball jackpot amounts and numbers of tickets sold, as shown in Section 10-1. Because the regression equation yn = -10.9 + 0.174x is a good model, substitute x = 625 into the regression equation to get a predicted value of 97.9 million tickets sold. The actual number of tickets sold was 90 million, so the predicted value of 97.9 million tickets is pretty good. b. Bad Model: Use y for predictions. There is no correlation between height and IQ score, so we know that a regression equation is not a good model. Therefore, the best predicted IQ score value is the mean IQ score, which is 100. INTERPRETATION Note that in part (a), the paired data result in a good regression model, so the predicted number of tickets sold is found by substituting the value of x = 625 into the regression equation. However, in part (b) there is no correlation between height and IQ, so the best predicted IQ score is the mean IQ score of y = 100. Key point: Use the regression equation for predictions only if it is a good model. If the regression equation is not a good model, use y for the predicted value of y. PART 2 Beyond the Basics of Regression In Part 2 we consider the concept of marginal change, which is helpful in interpreting a regression equation; then we consider the effects of outliers and special points called influential points. We also consider residual plots. Interpreting the Regression Equation: Marginal Change We can use the regression equation to see the effect on one variable when the other variable changes by some specific amount. DEFINITION In working with two variables related by a regression equation, the marginal change in a variable is the amount that it changes when the other variable changes by exactly one unit. The slope b1 in the regression equation represents the marginal change in y that occurs when x changes by one unit. C Machine Learning A relatively new and growing field is machine learning. Machine learning in a system (such as a self-driving car) uses artificial intelligence (AI) in a way that enables the system to learn from experience instead of direct human intervention. This new field requires the use of statistics, including topics such as descriptive statistics, outlier detection, data sampling, experimental design, determining when results are significant, normal distributions, correlation, confidence intervals, and hypothesis testing. Because these topics are included in this book, this book becomes a great beginning in the study of machine learning. A n g m in le s
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