All Notes on All Formulas/Tests PDF

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Summary

These are all the equations and "tests" that were covered in the course....

Description

μ = population mean 𝑥 =sample mean descriptive stats describe data (graphically, numerically)

inferential stats infer about population

Frequency table → frequency/bin ranges are equal; absolute, relative or cumulative frequency # of bins ≈ √#observations (should be about 5-20-ish) no relative frequency at μ0 → β(μ') = Φ(𝑧𝛼 + 0 ) Ha: μp0 → β(p') = Φ( Ha: p≠p0 → β(p') = Φ(

)

𝑝0 −𝑝′+𝑧𝛼 √𝑝0 (1−𝑝0 )/𝑛 √𝑝′(1−𝑝′)/𝑛

𝑝0 −𝑝′+𝑧𝛼/2√𝑝0 (1−𝑝0 )/𝑛 √𝑝′(1−𝑝′)/𝑛

Sample size for β(p') → 1-tailed test n ≥ (

𝑝−𝑝0 √𝑝0 (1−𝑝0 ) ⁄𝑛

)−Φ(

Ha: p Δ0 → β(Δ ') = Φ(𝑧𝛼 −

Δ′ −Δ0

Ha: μ1 – μ2 ≠ Δ0 → β(Δ ') = Φ(𝑧𝛼/2 −

)

𝜎 Δ′ −Δ0 𝜎

(round down) Δ0 −Δ′ 𝜎𝑥−𝑦

) − Φ (−𝑧𝛼/2 −

) → n1, n2 ≥

CI: 𝑥 − 𝑦 ± 𝑡𝛼/2,𝑣 √𝜎12 /𝑛1 + 𝜎22 /𝑛2 2

(𝜎12 +𝜎22 )(𝑧𝛼 +𝑧𝛽 ) (Δ′ −Δ0 )2

Ha: μ1 – μ2 < Δ0 → β(Δ ') = 1 – Φ(−𝑧𝛼 − Δ′ −Δ 𝜎

0

)

Δ′ −Δ0 𝜎

σ = 𝜎𝑥 −𝑦 = √𝜎12 /𝑛1 + 𝜎22 /𝑛2

)

Paired t-test: usually strong positive dependence; paired t-test variance and CI are smaller/narrower than 2 sample t-test n individuals from one population; results from 2 experiments (e.g. grades of same people in 2 different classes) 𝑑 −Δ 0 H 0 : μ D = Δ0 t= v = d.f. = n-1 CI: 𝑑 ± 𝑡𝛼/2,𝑣 𝑠𝐷 /√𝑛 𝑠𝐷 /√𝑛

n1, n2 large

Compare 2 population proportions: H0: p1 – p2 = 0

z=

𝑝1 −𝑝2

Ha: p1 – p2 > 0 → β(p1,p2) = Φ(

Ha: p1 – p2 ≠ 0 → β(p1,p2) = Φ(

𝑧𝛼 √𝑝 𝑞 (

H0: 𝜎12 = 𝜎22

2

𝑛1 𝑝1 +𝑛2 𝑝2 𝑛1 +𝑛2

𝜎

)

𝜎

)−Φ(

1 1 + )−(𝑝1 −𝑝2 ) 𝑛1 𝑛2

=

#𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑒𝑠 𝑡𝑜𝑡𝑎𝑙

Ha: p1 – p2 < 0 → β = 1 – Φ( 1 1 + )−(𝑝1 −𝑝2 ) 𝑛1 𝑛2

−𝑧𝛼 √𝑝 𝑞( 2

2

(𝑧𝛼 √(𝑝1 +𝑝2 )(𝑞1 +𝑞2 )/2 +𝑧𝛽 √(𝑝1 𝑞1 +𝑝2 𝑞2 ) ) 𝑑2

𝑠2

f = 𝑠12

Ha: 𝜎12 > 𝜎22 → f ≥ Fα,v

1 1 + )−(𝑝1 −𝑝2 ) 𝑛1 𝑛2

𝑧𝛼 √𝑝 𝑞(

Sample size for a certain β → n ≥ f-test (2 variances):

𝑝 =

√𝑝𝑞(1/𝑛1 +1/𝑛2 )

2

v = d.f. = (n1-1,n2-1)

Ha: 𝜎12 < 𝜎22 → f ≤ F1-α,v

𝜎

)

𝜎 = 𝜎𝑝1 −𝑝2 = √

−𝑧𝛼 √𝑝𝑞  (

1

𝑛1

1

𝑝1 𝑞1 𝑛1

+ 𝑛 )−(𝑝1 −𝑝2 ) 𝜎

2

+ )

𝑝2 𝑞2 𝑛2

CI: 𝑝1 − 𝑝2 ± zα/2 𝜎𝑝1 −𝑝2

⇒ replace zα with zα/2 for 2-tailed test

Ha: 𝜎12 ≠ 𝜎22 → f ≥ Fα/2,v or f ≤ F1-α/2,v...