Triola Elementary Statistics · 11th Edition
Chapter 10: Correlation & Regression
Every key formula from Triola Elementary Statistics Chapter 10, in one searchable page. Click a card to study it — worked examples included.
10.2 · CORRELATION
Formula 10-1
r = n∑xy − (∑x)(∑y)√[n∑x² − (∑x)²] · √[n∑y² − (∑y)²]
Formula 10-2 (z-scores)
r = ∑(zx · zy)n − 1
KEY NOTES
- −1 ≤ r ≤ 1: ±1 = perfect line.
- r unchanged by scale or swapping x, y.
- r measures LINEAR strength only.
- Requirements: random sample, straight-line scatterplot, no outliers.
WORKED EXAMPLE
Pizza/subway costs, n = 6:
r = 15.47√(16.37)(14.9825) = 0.988 (strong positive).
r = 15.47√(16.37)(14.9825) = 0.988 (strong positive).
10.5–10.6 · MULTIPLE REGRESSION
ŷ = b0 + b1x1 + b2x2 + … + bkxk
Find with software. Keep the model SMALL: drop variables that barely raise adjusted R².
Adjusted R²
adj. R² = 1 − [(n−1)(1−R²)]/[n−(k+1)]
Overall P-value
Tests H0: β1 = β2 = … = βk = 0.
WORKED EXAMPLE
R² = 0.675, n = 20, k = 2 ⇒
adj. R² = 1 − (19)(0.325)/17 = 0.637.
adj. R² = 1 − (19)(0.325)/17 = 0.637.
Modeling
Compare R² over linear, quadratic, logarithmic, exponential, power models — then THINK: reject predictions that are absurd (y = 2.77x² − 6.00x + 10.01 gave 337M people in 2020, but 671M in 1492!).
10.3 · REGRESSION EQUATION
ŷ = b0 + b1x
Least-squares line: b1 = slope, b0 = y-intercept.
Slope
b1 = n∑xy − (∑x)(∑y)n∑x² − (∑x)² = r · (sy/sx)
y-intercept
b0 = ŷ − b1x̄
Round b1, b0 to three significant digits.
WORKED EXAMPLE
Pizza (x) → subway fare (y):
ŷ = 0.0346 + 0.945x.
ŷ = 0.0346 + 0.945x.
10.4 · r² AND RESIDUALS
Coefficient of determination
r² = explained variationtotal variation = proportion of y variation explained
Partition
Total: ∑(y − ŷ)²; explained: ∑(ŷ − ŷ)²;
unexplained: ∑(y − ŷ)² = residual sum.
unexplained: ∑(y − ŷ)² = residual sum.
WORKED EXAMPLE
r = 0.988 ⇒ r² = 0.976:
97.6% of subway-fare variation is explained by pizza cost.
97.6% of subway-fare variation is explained by pizza cost.
10.4 · PREDICTION INTERVAL
Standard error of estimate
se = √∑y² − b0∑y − b1∑xyn − 2
Prediction interval
ŷ − E < y < ŷ + E
E = tα/2se√1 + 1/n + n(x0 − x̄)²/[n∑x² − (∑x)²]
E = tα/2se√1 + 1/n + n(x0 − x̄)²/[n∑x² − (∑x)²]
tα/2 has n − 2 df. Use only if the model fits.
WORKED EXAMPLE
x0 = $2.25 ⇒ ŷ = 2.16, se = 0.123:
E = (2.776)(0.122987)(1.2905606) = 0.441
95% PI: ($1.72, $2.60).
E = (2.776)(0.122987)(1.2905606) = 0.441
95% PI: ($1.72, $2.60).
10.2 · TESTING CORRELATION
H0: ρ = 0 vs H1: ρ ≠ 0
Test statistic
t = r√[(1 − r²)/(n − 2)], df = n − 2
P-value from Table A-3; or use |r| vs Table A-6.
WORKED EXAMPLE
r = 0.988, n = 6 ⇒
t = 0.988√[(1−0.988²)/4] = 12.793,
P = 0.00022 ≤ 0.05 ⇒ reject H0: linear correlation.
t = 0.988√[(1−0.988²)/4] = 12.793,
P = 0.00022 ≤ 0.05 ⇒ reject H0: linear correlation.
WATCH OUT!
Mistakes that cost points
- Correlation ≠ causation — r close to 0 does not prove “no relationship”
- Never extrapolate: predictions only inside the x range of the data
- ∑x² is NOT (∑x)² — square the values FIRST
- Round r to 3 decimals; b1, b0 to 3 significant digits
- Prediction interval ≠ confidence interval for the mean
- Outliers can single-handedly flip r — check the scatterplot
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