Reading and interpreting a line of best fit
The line is already drawn. Your job is to say what its slope and intercept mean in the language of the problem, and to use it to predict.
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ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.
A line of best fit for house price against size is \(\hat{y} = 145x + 32{,}000\), where \(x\) is square feet and \(\hat{y}\) is price in dollars.
By how many dollars does the model say the price rises for each additional square foot?
Same model: \(\hat{y} = 145x + 32{,}000\).
What price does it predict for a house of zero square feet?
A line of best fit for a plant’s height against weeks of growth is \(\hat{y} = 2.4x + 3\), with height in centimetres.
Predict the height at 5 weeks.
Six students recorded how long they studied and what they scored. The line of best fit is \(\hat{y} = 5x + 57\), where \(x\) is hours studied and \(\hat{y}\) is the predicted score.
(a) At zero
(b) Predict
(c) One more hour
Say it in words
One caution about that last answer
The data only ran from 1 to 6 hours. Predicting at 8 hours steps outside that range, which is called extrapolation, and the further you go the less the line is worth.
Push it far enough and it breaks visibly: at 10 hours the line predicts 107, on a test marked out of 100. The equation happily produces a number; the number stops meaning anything. ALEKS sometimes asks whether a prediction is reasonable, and this is what it is looking for.