1. **State the problem:** We have data for 8 male students showing hours spent on social media ($x$) and their IB Diploma points ($y$). We need to plot a scatter diagram, calculate means, plot the mean point, find the regression line, and draw it. 2. **Scatter diagram:** Plot points $(6,43), (15,33), (26,27), (12,36), (13,39), (40,17), (33,20), (23,33)$ on graph paper. - Use scale: 2 cm = 5 hours on $x$-axis, 2 cm = 10 points on $y$-axis. 3. **Calculate means:** - Mean of $x$: $$\bar{x} = \frac{6 + 15 + 26 + 12 + 13 + 40 + 33 + 23}{8} = \frac{168}{8} = 21$$ - Mean of $y$: $$\bar{y} = \frac{43 + 33 + 27 + 36 + 39 + 17 + 20 + 33}{8} = \frac{248}{8} = 31$$ 4. **Plot mean point $M$:** Plot $M = (21, 31)$ on the scatter diagram and label it. 5. **Find regression line equation:** - Formula for regression line $y$ on $x$ is: $$y = a + bx$$ where $$b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$$ and $$a = \bar{y} - b\bar{x}$$ Calculate sums: - $\sum (x_i - \bar{x})(y_i - \bar{y}) = (6-21)(43-31)+(15-21)(33-31)+(26-21)(27-31)+(12-21)(36-31)+(13-21)(39-31)+(40-21)(17-31)+(33-21)(20-31)+(23-21)(33-31)$ = $(-15)(12)+(-6)(2)+(5)(-4)+(-9)(5)+(-8)(8)+(19)(-14)+(12)(-11)+(2)(2)$ = $-180 -12 -20 -45 -64 -266 -132 +4 = -715$ - $\sum (x_i - \bar{x})^2 = (-15)^2 + (-6)^2 + 5^2 + (-9)^2 + (-8)^2 + 19^2 + 12^2 + 2^2$ = $225 + 36 + 25 + 81 + 64 + 361 + 144 + 4 = 940$ Calculate slope: $$b = \frac{-715}{940} \approx -0.7617$$ Calculate intercept: $$a = 31 - (-0.7617)(21) = 31 + 15.9957 = 46.9957 \approx 47.0$$ Regression line equation: $$y = 47.0 - 0.76x$$ 6. **Draw regression line:** On the scatter diagram, draw the line $y = 47.0 - 0.76x$ passing through the points. **Final answers:** - Mean hours $\bar{x} = 21$ - Mean points $\bar{y} = 31$ - Mean point $M = (21, 31)$ - Regression line: $$y = 47.0 - 0.76x$$