1. **Problem Statement:** We have data on 10 workers showing their age (Y) and days absent (X). We want to: (a) Set up a scatter diagram of Age (Y) vs Days absent (X). (b) Develop a simple linear regression model of Age (Y) on Days absent (X) using the least squares method. (c) Use the model to predict: i. Age of a worker absent for 12 days. ii. Days absent for a worker aged 590 years (likely a typo, but we proceed mathematically). 2. **Scatter Diagram:** Plot points with X-axis as Days absent and Y-axis as Age. Each point is (X, Y) from the table. 3. **Regression Model:** The simple linear regression model is: $$Y = a + bX$$ where $b$ is the slope and $a$ is the intercept. Formulas: $$b = \frac{n\sum XY - \sum X \sum Y}{n\sum X^2 - (\sum X)^2}$$ $$a = \frac{\sum Y - b \sum X}{n}$$ 4. **Calculate sums:** Given data: $X = [15,6,10,18,9,7,14,11,5,8]$ $Y = [27,61,37,23,46,58,29,36,64,40]$ Calculate: $\sum X = 15+6+10+18+9+7+14+11+5+8 = 103$ $\sum Y = 27+61+37+23+46+58+29+36+64+40 = 421$ $\sum XY = 15\times27 + 6\times61 + 10\times37 + 18\times23 + 9\times46 + 7\times58 + 14\times29 + 11\times36 + 5\times64 + 8\times40$ $= 405 + 366 + 370 + 414 + 414 + 406 + 406 + 396 + 320 + 320 = 3817$ $\sum X^2 = 15^2 + 6^2 + 10^2 + 18^2 + 9^2 + 7^2 + 14^2 + 11^2 + 5^2 + 8^2$ $= 225 + 36 + 100 + 324 + 81 + 49 + 196 + 121 + 25 + 64 = 1219$ 5. **Calculate slope $b$:** $$b = \frac{10 \times 3817 - 103 \times 421}{10 \times 1219 - 103^2} = \frac{38170 - 43363}{12190 - 10609} = \frac{-5193}{1581} \approx -3.283$$ 6. **Calculate intercept $a$:** $$a = \frac{421 - (-3.283) \times 103}{10} = \frac{421 + 338.05}{10} = \frac{759.05}{10} = 75.905$$ 7. **Regression equation:** $$Y = 75.905 - 3.283X$$ 8. **Predictions:** (i) For $X=12$ days absent, predict age: $$Y = 75.905 - 3.283 \times 12 = 75.905 - 39.396 = 36.509 \approx 37$$ (ii) For $Y=590$ years (likely a typo, but we calculate days absent $X$): $$590 = 75.905 - 3.283X$$ $$-3.283X = 590 - 75.905 = 514.095$$ $$X = \frac{-514.095}{3.283} \approx -156.6$$ Since days absent cannot be negative, this prediction is not meaningful for $Y=590$. **Final answers:** - Regression model: $Y = 75.905 - 3.283X$ - Predicted age for 12 days absent: 37 years - Predicted days absent for age 590: Not meaningful (negative value)