566 CHAPTER 10 Correlation and Regression 3. Think. The forecast result of 399 million in 2040 seems reasonable. (As of this writing, the latest figures from the U.S. Bureau of the Census use much more sophisticated methods to project that the U.S. population in 2040 will be 373 million.) However, there is considerable danger in making estimates for times that are beyond the scope of the available data. For example, the quadratic model suggests that in 1492, the U.S. population was 663 million, which is a result statisticians refer to as ridiculous. The quadratic model appears to be good for the available data (1800–2020), but other models might be better if it is necessary to make future population estimates. YOUR TURN. Do Exercise 5 “Landing on the Moon.” Interpreting R2 EXAMPLE 2 In Example 1, we obtained the value of R2 = 0.9995 for the quadratic model. Interpret that value as it relates to the predictor variable of year and the response variable of population size. YOUR TURN. Do Exercise 3 “Interpreting R2.” SOLUTION In the context of the year>population data from Table 10-7, the value of R2 = 0.9995 can be interpreted as follows: 99.95% of the variation in the population size can be explained by the quadratic regression equation (given in Example 1) that relates year and population size. COVID-19 Virus Pandemic EXAMPLE 3 As this was being written in 2020, the COVID-19 virus was sweeping the world. In the United States and many other countries, residents were instructed to stay at home. Restaurants, movie theaters, and many other nonessential businesses were ordered closed. The 2020 Olympics were postponed for a year, and many other professional sports were cancelled or postponed. Listed below are numbers of deaths in the United States resulting from the virus. The deaths occurred on consecutive days beginning with March 16, 2020, which was very early in the growth of the virus. Predicting future deaths becomes critical because those predictions could determine the distribution of important resources, such as hospital ventilators used in attempts to prevent deaths. Modeling the data became important as we all sought a return to normal life. Consequently, there were extensive and intensive efforts to develop and revise models. Use the listed data to find a mathematical model and use it to predict the number of deaths on the following day. How accurate is the prediction when compared to the actual number of 367 deaths that did occur? 19 22 48 59 53 126 73 203 225 264 YOUR TURN. Do Exercise 7 “CD Yields.” SOLUTION Use the days coded as 1, 2, 3, . . . . The quadratic and exponential models are very close with R2 values of 0.918817 and 0.918732, respectively. Given that populations tend to grow exponentially, we choose the exponential model of y = 15.111.34x2 as best, even though its R2 value is very slightly lower than that of the quadratic model. The prediction for the next day is 378 deaths (388 if using unrounded coefficients), which is close to the actual number of 367 deaths. The quadratic model yields a prediction of 329 deaths.
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