1. **Stating the problem:** We have the production data of chili peppers (cabai) in tons from 2001 to 2012. We need to find: a. The linear regression equation using the least squares method. b. The estimated production for the year 2017 and a conclusion. 2. **Formula and explanation:** The least squares regression line is given by: $$y = a + bx$$ where: - $y$ is the dependent variable (production), - $x$ is the independent variable (year, transformed), - $b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$ is the slope, - $a = \bar{y} - b\bar{x}$ is the intercept. We transform years to $x$ by setting $x = \text{year} - 2000$ to simplify calculations. 3. **Data transformation and calculations:** | Year | $x$ | Production $y$ (tons) | |------|-----|-----------------------| | 2001 | 1 | 861150 | | 2002 | 2 | 766572 | | 2003 | 3 | 762795 | | 2004 | 4 | 757399 | | 2005 | 5 | 732609 | | 2006 | 6 | 794931 | | 2007 | 7 | 802810 | | 2008 | 8 | 853615 | | 2009 | 9 | 965164 | | 2010 | 10 | 1048934 | | 2011 | 11 | 893124 | | 2012 | 12 | 964221 | Calculate sums: $$n = 12$$ $$\sum x = 1+2+\cdots+12 = 78$$ $$\sum y = 861150 + 766572 + \cdots + 964221 = 10063004$$ $$\sum x^2 = 1^2 + 2^2 + \cdots + 12^2 = 650$$ $$\sum xy = 1\times861150 + 2\times766572 + \cdots + 12\times964221 = 68629554$$ 4. **Calculate slope $b$:** $$b = \frac{12 \times 68629554 - 78 \times 10063004}{12 \times 650 - 78^2} = \frac{823554648 - 785000312}{7800 - 6084} = \frac{38554336}{1716} \approx 22477.5$$ 5. **Calculate intercept $a$:** $$\bar{x} = \frac{78}{12} = 6.5$$ $$\bar{y} = \frac{10063004}{12} \approx 838583.7$$ $$a = 838583.7 - 22477.5 \times 6.5 = 838583.7 - 146103.75 = 692479.95$$ 6. **Regression equation:** $$y = 692479.95 + 22477.5x$$ 7. **Estimate production for 2017:** For 2017, $x = 2017 - 2000 = 17$ $$y = 692479.95 + 22477.5 \times 17 = 692479.95 + 382117.5 = 1074597.45$$ 8. **Conclusion:** The estimated chili production in 2017 is approximately 1,074,597 tons, showing an increasing trend in production over the years.