Fitting a linear model to a scatter plot means drawing a line that best fits the points for posters and sign-ups. The trend is positive and linear, with sign-ups rising as posters increase. Using points (2,18) and (8,54), slope is (54-18)/(8-2)=36/6=6 sign-ups per poster. The intercept would predict sign-ups with zero posters, meaningful for base interest. A slope of 6 matches the trend's steepness and scale. Misconception: negative slope for positive data, but here it's clearly upward. Check direction, calculate slope from points, and assess intercept fit.

When modeling stopping distance versus speed, physics tells us that stopping distance increases with speed, suggesting a positive relationship. If a car traveling at 20 mph has a stopping distance of about 30 feet and at 40 mph it's about 70 feet, the slope is approximately (70-30)/(40-20) = 40/20 = 2 feet per mph. The y-intercept represents stopping distance at 0 mph, which should theoretically be 0, but the model gives -10, suggesting the linear approximation works best within the measured range. The equation y=2x-10 captures the positive trend, though the negative intercept indicates this model shouldn't be extrapolated to very low speeds. A common error is expecting the intercept to always be physically meaningful - linear models sometimes work well only within the data range. Always consider whether extrapolation makes sense for the context.

Fitting a linear model involves a line through scatter plot points to show trends like earning rates. The plot of pages read (x) and points (y) is positive and linear. Given points (10,45) and (30,85), slope is (85-45)/(30-10) = 40/20 = 2 points per page. Intercept isn't directly needed but would be found by extending to x=0. The slope of 2 matches as it fits the calculated rate and scale. Misconception: inverting rise/run for too-large slopes like 20. Check direction, estimate slope from points, then find intercept if required.