1. **Problem Statement:** We have experimental data for variables $x$ and $y$: | $x$ | 4.00 | 4.75 | 5.25 | 6.00 | 6.75 | 7.50 | 8.00 | |-----|------|------|------|------|------|------|------| | $y$ | 1.75 | 3.00 | 3.50 | 3.99 | 5.00 | 5.25 | 6.25 | We need to plot $y$ against $x$ and then: a) Measure the gradient (slope) of the straight line. b) Estimate the error in the gradient. c) Measure the $y$-intercept. d) Estimate the error on the $y$-intercept. 2. **Formula and Important Rules:** The data suggests a linear relationship $y = mx + c$ where $m$ is the gradient and $c$ is the $y$-intercept. The gradient $m$ is calculated by: $$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$$ The $y$-intercept $c$ is the value of $y$ when $x=0$. Error estimation in gradient and intercept can be done by considering the maximum and minimum slopes and intercepts from the plotted points, or by using the least squares method if data is fitted. 3. **Step-by-step Solution:** **a) Measure the gradient (slope):** Choose two points on the best-fit straight line through the data. For example, using the first and last points: $$m = \frac{6.25 - 1.75}{8.00 - 4.00} = \frac{4.5}{4} = 1.125$$ **b) Estimate the error in the gradient:** Estimate the maximum and minimum slopes by choosing points that give the steepest and shallowest lines through the data spread. For example, using points (4.75, 3.00) and (7.50, 5.25): $$m_{max} = \frac{5.25 - 3.00}{7.50 - 4.75} = \frac{2.25}{2.75} \approx 0.818$$ Using points (5.25, 3.50) and (6.75, 5.00): $$m_{min} = \frac{5.00 - 3.50}{6.75 - 5.25} = \frac{1.5}{1.5} = 1.0$$ The error in gradient $\Delta m$ can be approximated as half the difference between max and min slopes: $$\Delta m = \frac{|m_{max} - m_{min}|}{2} = \frac{|0.818 - 1.0|}{2} = 0.091$$ **c) Measure the $y$-intercept:** Using the equation $y = mx + c$, rearranged as $c = y - mx$, use a point and the gradient: Using point (4.00, 1.75): $$c = 1.75 - 1.125 \times 4.00 = 1.75 - 4.5 = -2.75$$ **d) Estimate the error on the $y$-intercept:** Calculate intercepts using max and min slopes: Using $m_{max} = 0.818$ and point (4.00, 1.75): $$c_{max} = 1.75 - 0.818 \times 4.00 = 1.75 - 3.272 = -1.522$$ Using $m_{min} = 1.0$ and point (4.00, 1.75): $$c_{min} = 1.75 - 1.0 \times 4.00 = 1.75 - 4.00 = -2.25$$ Error in intercept $\Delta c$ is half the difference: $$\Delta c = \frac{|-1.522 - (-2.25)|}{2} = \frac{0.728}{2} = 0.364$$ **Final answers:** - Gradient $m = 1.125 \pm 0.091$ - $y$-intercept $c = -2.75 \pm 0.364$ These values represent the slope and intercept of the best-fit line through the data with estimated errors based on data spread.