1. **Stating the problem:** We have data points for screen time $X$ and corresponding sleep hours $Y$. We want to find a mathematical model that relates $Y$ to $X$. 2. **Choosing the model:** Since sleep decreases as screen time increases, a linear model $Y = mX + b$ is a good starting point. 3. **Formula for linear regression:** The slope $m$ and intercept $b$ are given by: $$m = \frac{n\sum XY - \sum X \sum Y}{n\sum X^2 - (\sum X)^2}$$ $$b = \frac{\sum Y - m \sum X}{n}$$ where $n$ is the number of points. 4. **Calculate sums:** $\sum X = 1+2+\cdots+10 = 55$ $\sum Y = 8.5 + 8.2 + 7.8 + 7.5 + 7 + 6.5 + 6.1 + 5.6 + 5.2 + 4.7 = 66.1$ $\sum X^2 = 1^2 + 2^2 + \cdots + 10^2 = 385$ $\sum XY = 1\times8.5 + 2\times8.2 + 3\times7.8 + 4\times7.5 + 5\times7 + 6\times6.5 + 7\times6.1 + 8\times5.6 + 9\times5.2 + 10\times4.7 = 326.4$ 5. **Calculate slope $m$:** $$m = \frac{10 \times 326.4 - 55 \times 66.1}{10 \times 385 - 55^2} = \frac{3264 - 3635.5}{3850 - 3025} = \frac{-371.5}{825} \approx -0.4503$$ 6. **Calculate intercept $b$:** $$b = \frac{66.1 - (-0.4503) \times 55}{10} = \frac{66.1 + 24.7665}{10} = \frac{90.8665}{10} = 9.0867$$ 7. **Final linear model:** $$Y = -0.4503X + 9.0867$$ 8. **Interpretation:** For each additional hour of screen time, sleep decreases by about 0.45 hours. 9. **Summary:** The linear regression line fits the data well and can be used to predict sleep hours based on screen time.