1. **Problem Statement:** We have scores of 10 students in Mathematics (x) and Science (y). We need to plot the scatter graph, find the mean point M, find the regression line equation, plot it, and predict a science score for a math score of 88. 2. **Scatter Plot:** The points are (90,85), (38,60), (58,78), (85,70), (73,65), (82,71), (56,80), (73,90), (95,96), (80,85). 3. **Find the mean point M (\bar{x}, \bar{y}):** Calculate means: $$\bar{x} = \frac{90+38+58+85+73+82+56+73+95+80}{10} = \frac{730}{10} = 73$$ $$\bar{y} = \frac{85+60+78+70+65+71+80+90+96+85}{10} = \frac{780}{10} = 78$$ So, point M is (73, 78). 4. **Regression line equation:** The formula for the regression line of y on x is: $$y = a x + b$$ where $$a = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$$ $$b = \bar{y} - a \bar{x}$$ Calculate sums: $$\sum (x_i - \bar{x})(y_i - \bar{y}) = (90-73)(85-78)+(38-73)(60-78)+\ldots+(80-73)(85-78) = 2042$$ $$\sum (x_i - \bar{x})^2 = (90-73)^2+(38-73)^2+\ldots+(80-73)^2 = 1562$$ Calculate slope: $$a = \frac{2042}{1562} \approx 1.307$$ Calculate intercept: $$b = 78 - 1.307 \times 73 = 78 - 95.411 = -17.411$$ So, regression line: $$y = 1.307 x - 17.411$$ 5. **Plot the regression line:** This line can be drawn on the scatter graph. 6. **Prediction for x=88:** Substitute $x=88$ into the regression line: $$y = 1.307 \times 88 - 17.411 = 114.996 - 17.411 = 97.585$$ Expected science score is approximately 98. This completes all parts of the problem.