Summary Formula sheet 2nd exam Statistics (GEO2-2217)

Formula sheet Statistics exam 2

Deviations
2
𝑠𝑋 = Σ𝑑𝑥𝑑𝑥 / (𝑛 − 1)
2
𝑠𝑌 = Σ𝑑𝑦𝑑𝑦 / (𝑛 − 1)


Covariance
𝐶𝑜𝑣 = Σ𝑑𝑥𝑑𝑦 / (𝑛 − 1)


Coefficient of a linear association
𝑟 = 𝐶𝑜𝑣 / (𝑠𝑥𝑠𝑦) → eta2 ≥ r2


Student distribution
𝑟
𝑇= 2
· 𝑛−2 for H0: ρ = 0
1−𝑟



Spearman’s rho
𝑟𝑠
𝑇= · 𝑛−2 for H0: ρ𝑠 = 0
2
1 − 𝑟𝑠



Kendall’s tau
+ −
𝑘 −𝑘
𝑡𝑎𝑢 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑎𝑖𝑟𝑠

𝑍= |𝐾+ − 𝐾−| −1 for H0: τ = 0
𝑛(𝑛 − 1)(2𝑛 + 5) / 18



Linear regression
𝑌 = β0 + β1𝑋1 + 𝑒𝑟𝑟𝑜𝑟


Variation
2 2 2
Total variation: 𝑆𝑆𝑑 = 𝑆𝑆𝑒 + 𝑆𝑆𝑙 𝑆𝑆𝑑 = 𝑑1 + 𝑑2 + 𝑑𝑥 𝑑 = 𝑦 −𝑦
2 2 2
Explained variation: 𝑆𝑆𝑙 = (𝑙1 · #𝑜𝑏𝑠 𝑔1) + (𝑙2 · #𝑜𝑏𝑠 𝑔2) + (𝑙𝑥 · #𝑜𝑏𝑠 𝑔𝑥) 𝑙 =𝑦−𝑦
2 2 2
Residual variation: 𝑆𝑆𝑒 = 𝑒1 + 𝑒2 + 𝑒𝑥 𝑒 = 𝑦 −𝑦
2
𝑟 = 𝑆𝑆𝑙 / 𝑆𝑆𝑑


Significance of beta
σ𝑒 2
Exc prob: 𝑆𝐸(𝐵) = 2
= 𝑠𝑒 / Σ(𝑥 − 𝑥)
Σ(𝑥 − 𝑥)
2 2
2 Σ𝑒 Σ(𝑦−𝑦)
𝑠𝑒 = 𝑑𝑓𝑒
= 𝑛−2

𝑡 = 𝑏 / 𝑆𝐸(𝐵)
2
CI: undirected; β𝑙𝑖𝑚𝑖𝑡 = 𝑏 ± 𝑡1/2α · 𝑠𝑒 / Σ(𝑥 − 𝑥)
2
directed; β𝑙𝑖𝑚𝑖𝑡 = 𝑏 ± 𝑡α · 𝑠𝑒 / Σ(𝑥 − 𝑥)