# CPI Prediction Using Least Squares Model ## Table Data | Year | CPI | |------|----------| | 201 | 217.403 | | 2011 | 224.093 | | 2012 | 229.187 | | 2013 | 231. | | 2014 | 236.468 | | 2015 | 236.222 | | 2016 | 238.992 | | 2017 | 244.274 | | 2018 | 250.275 | | 2019 | 255.326 | ## Step-by-Step Solution ### 1. Assign Variables Let \( x \) be the number of years since 201: - 201: \( x = \) - 2011: \( x = 1 \) - ... - 2019: \( x = 9 \) ### 2. Apply Least Squares Formula The regression line is: \[ y = a + bx \] where: - \( y \) = CPI - \( x \) = years since 201 Formulas for \( a \) and \( b \): \[ b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \] \[ a = \frac{\sum y - b\sum x}{n} \] ### 3. Calculate Sums Let’s calculate: - \( n = 10 \) - \( \sum x = + 1 + 2 + ... + 9 = 45 \) - \( \sum x^2 = ^2 + 1^2 + ... + 9^2 = 285 \) - \( \sum y = 217.403 + 224.093 + 229.187 + 231.797 + 236.468 + 236.222 + 238.992 + 244.274 + 250.275 + 255.326 = 2364.037 \) Now, calculate \( \sum xy \): \[ \begin{align*} & \times 217.403 = \\ &1 \times 224.093 = 224.093 \\ &2 \times 229.187 = 458.374 \\ &3 \times 231.797 = 695.391 \\ &4 \times 236.468 = 945.872 \\ &5 \times 236.222 = 1181.11 \\ &6 \times 238.992 = 1433.952 \\ &7 \times 244.274 = 1709.918 \\ &8 \times 250.275 = 2002.2 \\ &9 \times 255.326 = 2297.934 \\ \end{align*} \] Sum: \[ \sum xy = + 224.093 + 458.374 + 695.391 + 945.872 + 1181.11 + 1433.952 + 1709.918 + 2002.2 + 2297.934 = 10948.844 \] ### 4. Calculate Slope (\( b \)) and Intercept (\( a \)) \[ b = \frac{10 \times 10948.844 - 45 \times 2364.037}{10 \times 285 - 45^2} \] \[ b = \frac{109488.44 - 106381.665}{285 - 2025} \] \[ b = \frac{3106.775}{825} = 3.768 \] \[ a = \frac{2364.037 - 3.768 \times 45}{10} \] \[ a = \frac{2364.037 - 169.56}{10} = \frac{2194.477}{10} = 219.448 \] ### 5. Predict CPI for 202 (\( x = 10 \)) \[ y = 219.448 + 3.768 \times 10 = 219.448 + 37.68 = 257.128 \] --- ## **Estimated CPI for April 202:** \[ \boxed{257.128} \] *(Rounded to three decimal places as requested.)*