SME
Grade 9 Math · Problem type
The least-squares line and r
Use this for “Predict the drinks sold at 62 °F from the line of best fit, and describe r.”
Chart
Least-squares line: ŷ = −0.59x + 82.19
Prediction for x of 62: ŷ = −0.59 × 62 + 82.19 = 45.61
r ≈ −0.9: a strong negative correlation
Residual = actual − predicted: 4 positive (above the line), 4 negative
Sum of squared residuals ≈ 86.83
The least-squares line makes the sum of the squared residuals as small as it can be
This is an example. Type your own values to start a new problem.
ŷ
=
−0.59
×
x
+
82.19
=
Formulas
ŷ
=
−0.59
×
x
+
82.19
45.61 = −0.59 × 62 + 82.19
Assumptions
•
A calculator gives the least-squares line: here ŷ ≈ −0.59x + 82.19.
•
r ≈ −0.90: the negative sign says the points fall; near −1 says they are close to a line.
•
Correlation is not causation: warm days bring other changes too.
Step-by-step
WE KNOW
x = 62 °F
FIND
predicted drinks sold (ŷ)
Step 1 · Find predicted drinks sold (ŷ)
Use ŷ = −0.59 × x + 82.19
Put
x
into
the
line:
slope
times
x
,
plus
the
intercept.
ŷ
=
−0.59
×
62
+
82.19
ŷ
=
−36.58
+
82.19
ŷ
=
45.61
→
about
46
drinks
Check
Put the numbers back in. Both sides should match.
45.61 = −0.59 × 62 + 82.19 ✓
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Scatter plots, correlation, residuals and lines of best fit
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Positive, negative or none?
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