1. **Problem Statement:** We are given data on typing experience (in years) and the number of typographical errors made by 12 typists. We need to find the linear regression equation to estimate errors based on experience. 2. **Data:** Experience (X): 8, 10, 7, 15, 9, 12, 10, 5, 3, 12, 2, 9 Errors (Y): 10, 4, 12, 0, 7, 2, 8, 10, 22, 6, 16, 9 3. **Formula for linear regression line:** $$y = a + bx$$ where $$b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$ $$a = \frac{\sum y - b \sum x}{n}$$ 4. **Calculate sums:** $$n=12$$ $$\sum x = 8+10+7+15+9+12+10+5+3+12+2+9 = 102$$ $$\sum y = 10+4+12+0+7+2+8+10+22+6+16+9 = 106$$ $$\sum x^2 = 8^2+10^2+7^2+15^2+9^2+12^2+10^2+5^2+3^2+12^2+2^2+9^2 = 1024$$ $$\sum xy = (8)(10)+(10)(4)+(7)(12)+(15)(0)+(9)(7)+(12)(2)+(10)(8)+(5)(10)+(3)(22)+(12)(6)+(2)(16)+(9)(9) = 686$$ 5. **Calculate slope $b$:** $$b = \frac{12 \times 686 - 102 \times 106}{12 \times 1024 - 102^2} = \frac{8232 - 10812}{12288 - 10404} = \frac{-2580}{1884} = -1.369$$ 6. **Calculate intercept $a$:** $$a = \frac{106 - (-1.369) \times 102}{12} = \frac{106 + 139.638}{12} = \frac{245.638}{12} = 20.47$$ 7. **Regression equation:** $$y = 20.47 - 1.369x$$ 8. **Interpretation:** The negative slope indicates that as typing experience increases, the number of errors decreases. 9. **Estimate errors for 12 years experience:** $$y = 20.47 - 1.369 \times 12 = 20.47 - 16.428 = 4.04$$ So, a typist with 12 years experience is expected to make about 4 errors. 10. **Reliability comment:** The model fits the data linearly but real-world factors may cause deviations. Predictions far from the data range should be treated cautiously. 11. **Practical interpretation:** The regression equation helps predict errors based on experience, showing that more experience generally reduces errors.