Formulae and Tables:
∑𝑥
𝑥̅ =
𝑛
𝑛 ∑ 𝑥 2 −(∑ 𝑥)
, 𝑠=√
𝑡=
𝑛1 +𝑛2 −2
̅̅̅̅
̅̅̅̅
𝑋1 −𝑋
2 −(𝜇1 −𝜇2 )
2
2
𝑛1
𝑛2
Paired t test: t =
𝑑̅ −(𝜇1 −𝜇2 )
,
𝑠𝑑
⁄
𝑛
√
Grand mean 𝑥̅ =
, 𝑡=
1
1
+
𝑛1 𝑛2
2
𝑠2 𝑠2
( 1+ 2)
𝑛1 𝑛2
𝑠2
𝑠2
( 1⁄𝑛 )2 ( 2⁄𝑛 )2
1 +
2
𝑛1 −1
𝑛2−1
𝑠𝑝 √
, 𝐷𝐹 ≅
𝑠
𝑠
√ 1+ 2
𝑥̅ −𝜇0
,
𝑡
𝜎
⁄ 𝑛
√
̅̅̅̅
̅̅̅̅
𝑋1 −𝑋
2 −(𝜇1 −𝜇2 )
𝑧=
𝑛(𝑛−1)
(𝑛1 −1)𝑠12 +(𝑛2 −1)𝑠22
𝑠𝑝 = √
2
=
𝑥̅ −𝜇0
𝑠
⁄ 𝑛
√
, DF : n-1
, DF: 𝑛1 + 𝑛2 − 2
,
∑𝑑
𝑑̅ = 𝑛 𝑖
2
𝑛 ∑ 𝑑𝑖2 − (∑ 𝑑𝑖 )
𝑠𝑑 = √
and
𝑛1 𝑥̅ 1 + 𝑛2 𝑥̅ 2 +⋯+𝑛𝑘 𝑥̅ 𝑘
𝑛(𝑛−1)
, 𝑛 = 𝑛1 + 𝑛2 + ⋯ +
𝑛1 + 𝑛2 +⋯+𝑛𝑘
𝑆𝑆𝑇𝑅 = 𝑛1 (𝑥̅1 − 𝑥̅ )2 + 𝑛2 (𝑥̅2 − 𝑥̅ )2 + ⋯ + 𝑛𝑘 (𝑥̅𝑘 − 𝑥̅ )2
𝑆𝑆𝐸 = (𝑛1 − 1)𝑠12 + (𝑛2 − 1)𝑠22 + ⋯ + (𝑛𝑘 − 1)𝑠𝑘2
𝑆𝑆𝑇𝑅
𝑆𝑆𝐸
𝑀𝑆𝑇𝑅
𝑀𝑆𝑇𝑅 =
𝑡=
𝑘−1
𝑀𝑆𝐸 =
𝐹=
𝑛−𝑘
𝑀𝑆𝐸
, DF = n-1
𝑛𝑘
~𝐹(𝑘 − 1, 𝑛 − 𝑘)
𝛽̂ − 𝛽
~ 𝑛 − (𝑘 + 1)
𝑆𝐸(𝛽̂ )
[𝛽̂ : Estimated regression coefficient 𝛽: Population coefficient, 𝑆𝐸(𝛽̂ ): Standard error of
estimated coefficient]
Logistic Regression: The marginal effect of quantitative 𝑥𝑘 variable:
𝑑𝑝
𝑑𝑥𝑘
= 𝛽𝑘 𝑝 (1 − 𝑝)
Alternatively, ME of 𝑥𝑘 at value x is: ME = P(Y = 1 | Xk= x+1) − P(Y = 1 | Xk = x)
Marginal effect of a binary dummy variable D: ME = P(Y = 1 | D=1, X) −P(Y = 1 | D=0, X)
Chi Square Tests: 𝜒 2 = ∑
(𝑂𝑖 −𝐸𝑖 )2
𝐸𝑖
[O: observed freq, E: expected freq]
[DF: (r-1)(c-1) for test of independence and k-1 for goodness of fit test]
𝑅𝑜𝑤 𝑇𝑜𝑡𝑎𝑙 × 𝐶𝑜𝑙𝑢𝑚𝑛 𝑇𝑜𝑡𝑎𝑙
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 =
, Expected frequency = np
𝐺𝑟𝑎𝑛𝑑 𝑇𝑜𝑡𝑎𝑙