Use a linear model to solve problems and interpret its slope and intercept in context

Learning goal: Use the equation of a linear model to solve problems about bivariate measurement data, and interpret the slope and the intercept in the units and context of the data. This is the interpret step of the GAISE model: turning a fitted line into statements about the situation.

What the model is. In the previous lesson a straight line was fitted to ten hypothetical bean plants, giving h = 1.5s + 12, where s is daily hours of sunlight and h is predicted mature height in centimeters. That equation is a linear model of the relationship. The word model is a reminder that the equation describes the trend of the data; it is not a law, and actual plants scatter around it. The model has two numbers, and each has a meaning in the context.

Interpreting the slope. The slope is 1.5, and its unit is centimeters per hour, because it is a change in height divided by a change in sunlight. In context: an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. Three parts of that sentence carry the meaning. "An additional hour" names the change in the horizontal quantity. "1.5 cm" names the predicted change in the vertical quantity. "Is associated with" states what the model can support. The sentence "each hour of sunlight makes a plant grow 1.5 cm" says more than the data can show, because the model describes association, not cause.

A slope interpretation must carry both units. "The slope is 1.5" tells a reader nothing about plants. "1.5 cm per hour of daily sunlight" tells them how much and of what. When the slope is negative, the sentence says "associated with a decrease": a slope of −1.2 cups per degree means each additional degree is associated with 1.2 fewer cups, not "a negative number of cups".

Interpreting the intercept. The intercept is 12 cm. It is the height the model predicts when s = 0, that is, for a plant receiving no direct sunlight. Whether that prediction means anything depends on the data. The plants in the experiment received between 2 and 10 hours; nobody grew a plant at 0 hours. So the intercept is where the line crosses the vertical axis, a feature of the model that fixes its position, and not a measured fact about sunless plants. A bean plant in darkness might not reach 12 cm at all. The honest interpretation is: "the model predicts 12 cm at zero hours of sunlight, but 0 hours is outside the range of the data, so this value should not be trusted as a real prediction."

Sometimes the intercept does sit inside the data and is meaningful, for example a starting length before any time has passed. The test is always the same: was the horizontal quantity ever near zero in the data? If so, the intercept is a prediction like any other; if not, it is a positioning number.

Predicting from the model. To predict the height of a plant that will receive 6.5 hours, substitute: h = 1.5 × 6.5 + 12 = 9.75 + 12 = 21.75 cm. The prediction is a typical height for that much sunlight, not a promise about one plant. In the previous lesson each of the ten measured plants fell within 1 cm of the line. That describes how far the measured plants sat from the line; it does not establish how far a new plant will land, because ten plants are too few to say how large a miss is likely, and a new plant can miss the line by more than any of the ten did. Reporting "the model predicts about 21.75 cm; the ten plants we measured were each within 1 cm of the line" is more honest than reporting the decimal alone, and it keeps the prediction and the evidence about scatter separate.

Substitution runs the other way too. To find how much sunlight is associated with a predicted height of 24 cm, solve 24 = 1.5s + 12: subtract 12 to get 1.5s = 12, then divide by 1.5 to get s = 8 hours. Check: 1.5 × 8 + 12 = 24. The step students skip is the check; do it every time.

The slope answers comparison questions without any substitution. The difference in predicted height between plants at 4 and 9 hours is (9 − 4) × 1.5 = 5 × 1.5 = 7.5 cm, because each hour adds 1.5 cm and there are five extra hours. Substituting both and subtracting gives 25.5 − 18 = 7.5 cm, the same answer, which confirms the slope reasoning.

Staying inside the data. A prediction at 14 hours of sunlight is 1.5 × 14 + 12 = 33 cm. The arithmetic is fine, but 14 hours is beyond the measured range of 2 to 10, and nothing in the data says the pattern stays straight. Plants might stop growing taller, or be harmed. Predicting outside the data range is called extrapolation, and a model's reliability there is unknown. State the range with the prediction: "within the measured range of 2 to 10 hours the model predicts about 1.5 cm more height per hour."

A negative-slope model. A second hypothetical data set: a cocoa stand at winter football games recorded the outdoor temperature in degrees Fahrenheit, from 20 to 60, and the cups sold. A line fitted to the scatter plot was c = 80 − 1.2T, where T is temperature and c is predicted cups sold. The slope is −1.2 cups per degree: each additional degree of temperature is associated with 1.2 fewer cups sold. The intercept, 80 cups, is the predicted sales at 0 degrees, which is colder than any game in the data, so it positions the line rather than forecasting a real day. At 35 degrees the model predicts 80 − 1.2 × 35 = 80 − 42 = 38 cups. Sales of 20 cups are associated with 1.2T = 60, so T = 50 degrees. A ten-degree warming is associated with 12 fewer cups, because 10 × 1.2 = 12.

The misconception. Slope and intercept are not interchangeable with "the answer." A student asked what the model says about sunlight and height who replies "12 and 1.5" has read the equation but not interpreted it. The interpretation is the sentence that puts each number back into the situation with its unit and its condition: 1.5 cm of predicted height per hour of daily sunlight, within 2 to 10 hours, as an association; and 12 cm as the line's value at zero hours, outside the data.

Worked prediction with a check on range: How tall does h = 1.5s + 12 predict a plant with 3.5 hours of sunlight will be? 3.5 is inside the measured range of 2 to 10 hours, so the prediction is reasonable to use. h = 1.5 × 3.5 + 12 = 5.25 + 12 = 17.25 cm. The model's prediction is 17.25 cm. Separately, the measured plants were each within 1 cm of the line; that describes the data, and these ten plants alone do not establish how far a new plant will differ from 17.25 cm.

Worked solve for the horizontal quantity: For what temperature does c = 80 − 1.2T predict 32 cups? Set 32 = 80 − 1.2T. Subtract 80 from both sides: −48 = −1.2T. Divide by −1.2: T = 40 degrees. Check: 80 − 1.2 × 40 = 80 − 48 = 32. Forty degrees is inside the recorded range of 20 to 60, so the answer is within the data.

Worked interpretation of both parameters: A hypothetical study of used bicycles fits price in dollars against age in years and gets p = 260 − 30a for bicycles aged 1 to 6 years. Slope: each additional year of age is associated with a price 30 dollars lower. Intercept: 260 dollars is the model's predicted price at age 0, a new bicycle; since the youngest bicycle in the data was 1 year old, the intercept is just outside the data, and it should be described as the line's starting value rather than a measured new-bicycle price. Predicted price at 4 years: 260 − 120 = 140 dollars. The model predicts 0 dollars at age 260 ÷ 30, about 8.7 years, which is beyond the data and probably not true of real bicycles; the model is not trusted there.

Interpretation cards: For each of the three models in this lesson, h = 1.5s + 12, c = 80 − 1.2T and p = 260 − 30a, write a card with four lines: the slope with its unit and its "associated with" sentence; the intercept with its unit and whether zero is inside the data range; one prediction inside the range with a check; and one question that would require extrapolation, with a sentence saying why the answer is uncertain.

Independent performance: Take the fitted line and equation from your own data in the previous lesson. Write an interpretation of its slope in a full sentence with both units, and of its intercept with a judgement about whether zero was inside your data. Solve three problems with the model: predict the vertical quantity for a chosen horizontal value inside your range, solve for the horizontal value that gives a chosen vertical value and check it by substitution, and find the predicted difference between two horizontal values using the slope alone. Finish with one sentence on the largest gap you observed in the previous lesson and how much an actual value might differ from your prediction. Success means every number carries a unit, every equation is checked, the word "associated" rather than "causes" is used, and the data range is stated.