Summary of Formulas/Tests (Chapter 10&12)
Chapter 10: Correlation and Regression
1. Linear Correlation Coefficient
𝑛(∑ 𝑥𝑦) − (∑ 𝑥)(∑ 𝑦)
𝑟=
√𝑛(∑ 𝑥²) − (∑ 𝑥)² √𝑛(∑ 𝑦²) − (∑ 𝑦)²

Notation:
n: the number of pairs of data present
r: the linear correlation coefficient for a sample, -1 ≤ r ≤ +1
If r is close to 0, we conclude that there is no linear correlation between x and y,
but if r is close to -1 or +1, we conclude that there is a linear correlation
between x and y.
2. Coefficient of Determination: r²
The value of r² is the proportion of the variation in y that is explained by the linear
relationship between x and y.
3. Hypothesis Test for Correlation
H0: ρ = 0
(There is no linear correlation.)
H1: ρ ≠ 0
(There is a linear correlation.)
(ρ: Greek letter rho used to represent the linear correlation coefficient for a population.)

Method 1: Test statistics is t:
Test statistic: 𝑡 =
𝑟
√1−𝑟²
𝑛−2
Critical values: t distribution (Table A-3 with df = n – 2)

Method 2: test statistic is r
Test statistic: 𝑟 =
𝑛(∑ 𝑥𝑦)−(∑ 𝑥)(∑ 𝑦)
√𝑛(∑ 𝑥²)−(∑ 𝑥)²√𝑛(∑ 𝑦²)−(∑ 𝑦)²
Critical value: critical values of the person correlation coefficient r (Table A-6)
4. Regression Equation
 Notations for regression equation
Population parameter
y-intercept of regression equation
β0
Slope of regression equation
β1
Equation of the regression line
Y = β0 + β1x
b1=
Sample statistic
b0
b1
𝑦̂ = b0 + b1x
𝑛(∑ 𝑥𝑦)−(∑ 𝑥)(∑ 𝑦)
𝑛(∑ 𝑥²)−(∑ 𝑥)²
b0= 𝑦 − b1𝑥 or 𝑏0 =
(∑ 𝑦)(∑ 𝑥 2 )−(∑ 𝑥)(∑ 𝑥𝑦)
𝑛((∑ 𝑥 2 )−(∑ 𝑥)²
𝑦̂ = b0 + b1x
5.
6.
7.
8.
Residual = observed y – predicted y = y - 𝑦̂
Total Deviation = y - 𝑦, total variation = ∑(𝑦 − 𝑦)²
Explained Deviation = y - 𝑦̂ − 𝑦, explained variation = ∑( 𝑦̂ − 𝑦)²
Unexplained Deviation = 𝑦 − 𝑦̂, unexplained variation = ∑(𝑦 − 𝑦̂)²
9. Coefficient of Determination: 𝑟² =
10. Standard Error of Estimate: se= √
𝑒𝑥𝑝𝑙𝑎𝑖𝑛𝑒𝑑 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛
𝑡𝑜𝑡𝑎𝑙 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛
∑(𝑦−𝑦̂)²
𝑛−2
or se= √
1
𝑛
+
− 𝑦)²
∑ 𝑦²−𝑏0 ∑ 𝑦−𝑏1 ∑ 𝑥𝑦
11. Prediction Interval for an Individual y
Margin of error: E = tα/2Se√1 +
∑(𝑦̂−𝑦)²
= ∑(y
𝑛(𝑥0 −𝑥)²
𝑛(∑ 𝑥²)−(∑ 𝑥)²
Prediction interval: 𝑦̂ − 𝐸 < 𝑦 < 𝑦̂ + 𝐸
X0: given value of x
tα/2: t distribution (Table A-3 with df = n-2)
12. Multiple Regression
Multiple regression equation: 𝑦̂ = b0 + b1x1 + b2x2 + ⋯ + bkxk
𝑛−2
Adjusted coefficient of determination: 𝑅² = 1 −
(𝑛−1)
[𝑛−(𝑘+1)]
(1 − 𝑅2 )
n: sample size
k: number of predictor (x) variables
Chapter 12: Analysis of Variance
1. One-Way ANOVA with Equal Sample Sizes n
H0: µ1 = µ2 = µ3 = ...
H1: At least one of the means is different from the others.
Test statistic: 𝐹
=
𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑎𝑚𝑝𝑙𝑒𝑠
𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑤𝑖𝑡ℎ𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒𝑠
=
𝑛𝑠𝑥2
𝑠𝑝2
=
𝑛𝑠𝑥2
2
2
𝑠2
1 +𝑠2 +⋯+𝑠𝑘
𝑘
Critical values: F distribution (Table A-5 with numerator df = k – 1 and denominator
df = k (n - ), k = number of samples and n = sample size)
2. One-Way ANOVA with Unequal Sample Sizes
H0: µ1 = µ2 = µ3 = ...
H1: At least one of the means is different from the others.
Test statistic:
𝐹=

𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑎𝑚𝑝𝑙𝑒𝑠
𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑤𝑖𝑡ℎ𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒𝑠
=
∑ 𝑛𝑖 (𝑥𝑖−𝑥
̿)²
]
𝑘−1
∑(𝑛 −1)𝑠2
[ ∑(𝑛𝑖 −1)𝑖 ]
𝑖
[
=
𝑀𝑆 (𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡)
𝑀𝑆 (𝑒𝑟𝑟𝑜𝑟)
Notation:
𝑥̿ : mean of all sample values combined
𝑘: number of population means being compared
𝑛𝑖 : number of values in the ith sample
𝑥𝑖 : mean of values in the ith sample
𝑠𝑖2 : variance of values in the ith sample
Critical values: F distribution (Table A-5 with numerator df = k – 1 and denominator
df = N – k with k = number of samples and N = total number of values in all samples
combined)
3. Two-Way ANOVA
H0: There are no effects from the row factor (that is, the row means are equal).
H1: There are no effects from the column factor (that is, the column means are equal).
Step 1. Interaction Effect
Test for an interaction between the two factors using 𝐹 =
𝑀𝑆 (𝑖𝑛𝑡𝑒𝑟𝑎𝑐𝑡𝑖𝑜𝑛)
𝑀𝑆(𝑒𝑟𝑟𝑜𝑟)
Step 2. Row/Column Effect
Is there an effect due to interaction between the two factors?
 Yes (Reject H0 of no interaction effect.). Stop. Don’t consider the effects
of either factor without considering the effects of the other.
 No (Fail to reject H0) of no interaction effect.).
 Test for effect from row factor using 𝐹 =
𝑀𝑆 (𝑟𝑜𝑤 𝑓𝑎𝑐𝑡𝑜𝑟)
𝑀𝑆(𝑒𝑟𝑟𝑜𝑟)
Interpretation: Compare P-value with significance level. If the P-value
is less than or equal to the significance level, reject the null
hypothesis of no effects from the row factor. If the P-value is greater
than the significance level, fail to reject the null hypothesis of no
effects from the row factor.
 Test for effect from column factor using 𝐹 =
𝑀𝑆 (𝑐𝑜𝑙𝑢𝑚𝑛 𝑓𝑎𝑐𝑡𝑜𝑟)
𝑀𝑆(𝑒𝑟𝑟𝑜𝑟)
Interpretation: Compare P-value with significance level. If the P-value
is less than or equal to the significance level, reject the null
hypothesis of no effects from the column factor. If the P-value is
greater than the significance level, fail to reject the null hypothesis of
no effects from the column factor.