Mathematics
Grade 9
15 min
Interpret regression lines
Interpret regression lines
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1
Introduction & Learning Objectives
Learning Objectives
Identify a regression line on a scatter plot.
Describe the general direction of a trend shown by a regression line (positive, negative, or no clear trend).
Estimate values for one variable given another, using a regression line.
Explain what a point on a regression line represents in a real-world context.
Recognize that regression lines show a general relationship, not a perfect fit for every data point.
Use rational numbers (decimals and fractions) when reading and interpreting values from a regression line.
Differentiate between the actual data points and the predicted values on the regression line.
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Key Concepts & Vocabulary
TermDefinitionExample
Scatter PlotA graph that shows the relationship between two sets of data using individual points.A scatter plot showing the age of a car and its selling price, with each point representing one car.
Data PointA single point on a scatter plot that represents two related values from the data set.On a graph of height vs. weight, a data point at (150 cm, 45 kg) means one person is 150 cm tall and weighs 45 kg.
TrendA general pattern or direction that the data points on a scatter plot seem to follow.If points generally go up from left to right, there's an 'increasing trend'.
Regression Line (Line of Best Fit)A straight line drawn through the middle of the data points on a scatter plot that best shows the overall trend or relationship between the two variable...
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Core Formulas
Interpreting Positive Trend
$x \uparrow \implies y \uparrow$
If the regression line slopes upwards from left to right, it indicates a positive trend. This means that as the value on the horizontal (x) axis increases, the predicted value on the vertical (y) axis also tends to increase.
Interpreting Negative Trend
$x \uparrow \implies y \downarrow$
If the regression line slopes downwards from left to right, it indicates a negative trend. This means that as the value on the horizontal (x) axis increases, the predicted value on the vertical (y) axis tends to decrease.
Estimating Values from the Line
Given $x_1$, find $y_1$ on the line.
To estimate a value for 'y' given an 'x' value, locate $x_1$ on the horizontal axis, move vertically up or down to th...
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Challenging
A scientist is studying the relationship between the depth of a lake and the water temperature. As the depth increases, the temperature decreases. Which of the following statements best describes the regression line for this data?
A.The regression line will have a positive slope.
B.The regression line will have a negative slope.
C.The regression line will be a flat, horizontal line.
D.It is impossible to draw a regression line for this data.
Challenging
A regression line predicts the number of customers in a store (y-axis) based on the hour of the day (x-axis, from 1 to 8). The line passes through (2, 30) and (5, 75). According to the line, an actual measurement at hour 3 showed 15 fewer customers than predicted. How many customers were actually in the store at hour 3?
A.30
B.45
C.50
D.60
Challenging
A regression line is created for the price of a pizza (y-axis) based on its diameter in inches (x-axis). The line passes through (8, 9.50) and (16, 17.50). A student concludes, 'Based on the line, a 24-inch pizza should cost exactly $25.50.' Why might this be an unreliable conclusion?
A.The student's math is wrong; it should be $26.50.
B.Pizza prices are always random and cannot be predicted.
C.The linear trend may not continue for much larger pizzas; pricing might change for specialty sizes.
D.The data points for 8-inch and 16-inch pizzas are probably wrong.
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