 = level of significance = type 1 error
 = type 2 error
Cheat Sheet of Doom and Despair
Power = 1-
1. Hypothesis on m when s is known
𝑍0 =
𝑋̄−𝜇0
𝜎 ⁄√𝑛
.
reject H0 if: Z0 > |Za/2| or Z0 < -|Za/2|
If H1:  > 0
reject H0 if Z0 > |Z|.
If H1:  < 0
reject H0 if Z0 < |Z|
P-Values (reject if p-value <
)
H1: m ¹ m0 à 𝑃 = 2𝑃(𝑍 > |𝑍0 |)
H1: m > m0 à P = P(Z > Z0)
H1: m < m0 à P = P(Z < Z0)
Type 2 Error (beta)
 = 0 + 
2 Sided: 𝛽 = 𝑃 (−𝑍𝛼/2 −
𝛿√𝑛
𝜎
≤ 𝑍 ≤ 𝑍𝛼/2 −
If H1: m > m0: 𝛽 = 𝑃 (𝑍 ≤ 𝑍𝛼 −
𝛿√𝑛
𝜎
𝛿√𝑛
𝜎
)
)
If H1: m < m0: 𝛽 = 𝑃 (𝑍 ≥ −𝑍𝛼 −
𝛿√𝑛
𝜎
)
Sample Size ( + 𝛽 known)
2
𝑛=
2
(𝑍𝛼/2 +𝑍𝛽 ) 𝜎 2
If H1 is one sided: 𝑛 =
𝛿2
(𝑍𝛼 +𝑍𝛽) 𝜎 2
𝛿2
Confidence Intervals
Upper: 𝑃 (𝜇 ≤ 𝑋̄ + 𝑍𝛼
𝜎
√𝑛
Lower: 𝑃 (𝜇 ≥ 𝑋̄ − 𝑍𝛼
)=1−𝛼
Half-Width (E)- 𝐸 = 𝑍𝛼/2
𝜎
Two-Sided: 𝑛 = (
√𝑛
𝜎
√𝑛
)=1−𝛼
𝑍𝛼/2 𝜎 2
𝑍 𝜎 2
𝐸
𝐸
) One Sided: 𝑛 = ( 𝛼 )
2. Hypothesis on  when  is unknown (T-test)
𝑇0 =
𝑋̄−𝜇0
reject H0 if: T0 > ta/2,n-1
𝑆⁄√𝑛
If H1:  > 0
reject H0 if: T0 > t,n-1
If H1:  < 0
reject H0 if: T0 < t,n-1
or
T0 < -ta/2,n-1.
DOF = n - 1
P-Values (reject if p-value <

)
H1: m ¹ m0 à P = 2 P t n 1  T0

H1: m > m0 à P = P(tn-1 > T0)
H1: m < m0 à P = P(tn-1 < T0)
Type 2 Error and Sample Size/ OC Function (graphs)
𝑑=
|𝜇 − 𝜇0 |
𝜎
Confidence Intervals
Upper: 𝑃 (𝜇 ≤ 𝑋̄ + 𝑡𝛼,𝑛−1
𝑆
√𝑛
) = 1−𝛼
Lower: 𝑃 (𝜇 ≥ 𝑋̄ − 𝑡𝛼,𝑛−1
𝑆
√𝑛
)= 1−𝛼
 = level of significance = type 1 error
 = type 2 error
Cheat Sheet of Doom and Despair
Power = 1-
If H1:   0, construct a 100(1  )% two sided confidence interval on .
If H1:  > 0, construct a 100(1  )% lower-confidence interval on .
If H1:  < 0, construct a 100(1  )% upper-confidence interval on .
3. Hypothesis on the Variance (Chi-Square)
𝜒02 =
reject H0 if:  0    / 2, n 1 or if  0   1 / 2, n 1
(𝑛−1)𝑆 2
2
𝜎02
2
2
If H1: 
2
  02
reject H0 if:  0    , n 1
If H1: 
2
  02
reject H0 if:  0   1 , n 1
2
H1: 𝜎
2
2
2
P-Values (reject if p-value <
≠ 𝜎02
2
2
)
2
2
𝑃 = 2 𝑚𝑖𝑛{𝑃(𝜒𝑛−1
> 𝜒02 ), 𝑃(𝜒𝑛−1
< 𝜒02 )}
H1: 𝜎 2 > 𝜎02
2
𝑃 = 𝑃(𝜒𝑛−1
> 𝜒02 )
H1: 𝜎 2 < 𝜎02
2
𝑃 = 𝑃(𝜒𝑛−1
< 𝜒02 )
Type 2 Error and Sample Size/ OC Function (graphs)
𝜆=
𝜎
𝜎0
Confidence Intervals
Upper: 𝑃 (𝜎 2 ≤
(𝑛−1)𝑆 2
2
𝜒1−𝛼,𝑛−1
Lower: 𝑃 (𝜎 2 ≥
)= 1−𝛼
(𝑛−1)𝑆 2
2
𝜒𝛼,𝑛−1
)=1−𝛼
4. Population Proportions
H0: p = p0
H0: p = p0
H0: p = p0
H1: p  p0
H1: p > p0
H1: p < p0
𝑍0 =
𝑋−𝑛𝑝0
reject H0 if: Z0 > |Za/2| or if Z0 < -|Za/2|
√𝑛𝑝0 (1−𝑝0 )
If H1: p > p0
reject H0 if: Z0 > |Z |
If H1: p < p0
reject H0 if Z0 < |Z |
P-Values

H1: p ¹ p0 à P  2P Z  Z 0
 H : p > p à P = P(Z > Z )
1
0
0
Type 2 Error/OC Function
2-sided: 𝛽 = 𝑃 (
𝑝 (1−𝑝 )
𝑝0 −𝑝−𝑍𝛼 √ 0 𝑛 0
2
√𝑝(1−𝑝)
𝑛
≤𝑍≤
𝑝 (1−𝑝 )
𝑝0 −𝑝+𝑍𝛼 √ 0 𝑛 0
2
√𝑝(1−𝑝)
𝑛
)
H1: p < p0 à P = P(Z < Z0)
 = level of significance = type 1 error
 = type 2 error
Cheat Sheet of Doom and Despair
Power = 1-
If H1: p > p0
𝛽 = 𝑃 (𝑍 ≤
𝑝 (1−𝑝0 )
𝑛
𝑝0 −𝑝+𝑍𝛼 √ 0
√
𝑝(1−𝑝)
𝑛
)
If H1: p < p0
𝛽= 𝑃 (𝑍 ≥
𝑝 (1−𝑝0 )
𝑛
𝑝0 −𝑝−𝑍𝛼 √ 0
√
𝑝(1−𝑝)
𝑛
)
Sample Size to achieve a certain power
𝑍𝛼/2 √𝑝0 (1−𝑝0 )+𝑍𝛽 √𝑝(1−𝑝)
two-sided test
𝑛=(
one-sided test
𝑛=(
2
)
𝑝−𝑝0
𝑍𝛼 √𝑝0 (1−𝑝0 )+𝑍𝛽 √𝑝(1−𝑝)
2
)
𝑝−𝑝0
𝑋
Confidence Intervals (𝑝̂ = )
𝑛
𝑝̂(1−𝑝̂)
Upper: 𝑃 (𝑝 ≤ 𝑝̂ + 𝑍𝛼 √
𝑝̂(1−𝑝̂)
Lower: 𝑃 (𝑝 ≥ 𝑝̂ − 𝑍𝛼 √
)=1−𝛼
𝑛
𝑛
)=1−𝛼
Sample Size w/E
2
two-sided
Z

n    / 2  p1  p 
 E 
2
Z

n    / 2  0.25
 E 
2
one-sided
Two Populations
1. Hypothesis on two  when  is known
𝑍0 =
(𝑋̄1 −𝑋̄2 )−𝛥0
reject H0 if: Z0 > Za/2 or if Z0 < -Za/2
𝜎2 𝜎2
𝑛1 𝑛2
√ 1+ 2
If H1: 1  2 > 0
reject H0 if: Z0 > Z
If H1: 1  2 < 0
reject H0 if: Z0 < Z
P-Values

H1: m1 - m2 ¹ D0 à P  2P Z  Z 0

H1: m1 - m2 > D0 à P = P(Z > Z0)
H1: 1  2 < 0  P = P(Z < Z0)
Type 2 Error/OC Function ( = 1  2)
2-sided: 𝛽 = 𝑃 −𝑍𝛼/2 −
(
If H1: m1 - m2 > D0
𝛥−𝛥0
𝜎2 𝜎2
√ 1+ 2
𝑛1 𝑛2
≤ 𝑍 ≤ 𝑍𝛼/2 −
𝛽 = 𝑃 𝑍 ≤ 𝑍𝛼 −
𝛥−𝛥0
𝜎2 𝜎2
𝑛1 𝑛2
√ 1+ 2
𝛥−𝛥0
𝜎2 𝜎2
𝑛1 𝑛2
√ 1+ 2
(
)
)
Z 
n     0.25
 E 
 = level of significance = type 1 error
 = type 2 error
Cheat Sheet of Doom and Despair
Power = 1-
𝛥−𝛥0
𝛽 = 𝑃 𝑍 ≥ −𝑍𝛼 −
If H1: m1 - m2 < D0
𝜎2 𝜎2
𝑛1 𝑛2
√ 1+ 2
(
)
Sample Size to achieve a certain power (n = n1 = n2)
2
𝑛=
two-sided test
2
(𝑍𝛼/2 +𝑍𝛽 ) (𝜎12 +𝜎22 )
one-sided test 𝑛 =
(𝛥−𝛥0 )2
(𝑍𝛼 +𝑍𝛽) (𝜎12 +𝜎22 )
(𝛥−𝛥0 )2
Using OC Curves for Computing  and Sample Size
𝑑=
|𝛥−𝛥0 |
√𝜎12 +𝜎22
H1: 1  2  0 use Charts VII(a) and VII(b)
H1: 1  2 > 0 use Charts VII(c) and VII(d)
H1: 1  2 < 0 use Charts VII(c) and VII(d)
Confidence Intervals
𝜎2
𝜎2
𝑛1
𝑛2
𝜎2
𝜎2
𝑛1
𝑛2
Upper
𝑃 (𝜇1 − 𝜇2 ≤ (𝑋̄1 − 𝑋̄2 ) + 𝑍𝛼 √ 1 + 2 ) = 1 − 𝛼
Lower
𝑃 (𝜇1 − 𝜇2 ≥ (𝑋̄1 − 𝑋̄2 ) − 𝑍𝛼 √ 1 + 2 ) = 1 − 𝛼
Choice of Sample Size for Confidence Intervals
2-sided: 𝑛 = (
𝑍𝛼/2 2
𝐸
𝑍
) (𝜎12 + 𝜎22 )
2
1-sided: 𝑛 = ( 𝛼) (𝜎12 + 𝜎22 )
𝐸
2. Hypothesis on two  when  is unknown (Unknown but EQUAL)
𝑆𝑝2 =
(𝑛1 −1)𝑆12 +(𝑛2 −1)𝑆22
𝑛1 +𝑛2 −2
𝑇0 =
(𝑋̄1 −𝑋̄2 )−𝛥0
𝑆𝑝 √
1
1
+
𝑛1 𝑛2
n = n1 + n2 - 2
H1: 1  2  0
reject H0 if: T0 > t/2, or if T0 < t/2,
H1: 1  2 > 0
reject H0 if: T0 > t,
H1: 1  2 < 0
reject H0 if: T0 < t,
P-Value

H1: m1 - m2 ¹ D0 à P  2P t  T0

H1: m1 - m2 > D0 à P = P(tn > T0)
H1: 1  2 < 0  P = P(t < T0)
Using OC Curves for Computing  and Sample Size
 = level of significance = type 1 error
 = type 2 error
Cheat Sheet of Doom and Despair
Power = 1-
D = m1 - m2
𝑑=
n = n1 = n2
|𝛥−𝛥0 |
2𝜎
sample sizes n* on the OC curves are actually 2n  1
two-sided test
use Charts VII(e) and VII(f)
one-sided test
use Charts VII(g) and VII(h)
𝑛=
𝑛∗ + 1
2
Confidence Intervals
Upper: 𝑃 (𝜇1 − 𝜇2 ≤ (𝑋̄1 − 𝑋̄2 ) + 𝑡𝛼,𝜈 𝑆𝑝 √
1
Lower: 𝑃 (𝜇1 − 𝜇2 ≥ (𝑋̄1 − 𝑋̄2 ) − 𝑡𝛼,𝜈 𝑆𝑝 √
1
𝑛1
𝑛1
+
+
1
𝑛2
1
𝑛2
)= 1−𝛼
)= 1−𝛼
3. Hypothesis on two  when  is unknown and UNEQUAL
𝑇0 =
(𝑋̄1 −𝑋̄2 )−𝛥0
𝑆2 𝑆2
𝑛1 𝑛2
2
𝑆2 𝑆2
𝑛1 𝑛2
2
2
2
(𝑆2
1 ⁄𝑛1 ) +(𝑆2 ⁄𝑛2 )
𝑛1 +1
𝑛2 +1
( 1+ 2)
𝜈=
√ 1+ 2
−2
(round down)
H1: 1  2  0
reject H0 if T0 > t/2, or if T0 < t/2,
H1: 1  2 > 0
reject H0 if T0 > t,
H1: 1  2 < 0
reject H0 if T0 < t,
P-Values

H1: m1 - m2 ¹ D0 à P  2P t  T0

H1: m1 - m2 > D0 à P = P(tn > T0)
H1: 1  2 < 0  P = P(t < T0)
Confidence Intervals
2
2
𝑛1
𝑛2
𝑆2
𝑆2
𝑛1
𝑛2
𝑆
𝑆
Upper: 𝑃 (𝜇1 − 𝜇2 ≤ (𝑋̄1 − 𝑋̄2 ) + 𝑡𝛼,𝜈 √ 1 + 2 ) = 1 − 𝛼
Lower: 𝑃 (𝜇1 − 𝜇2 ≥ (𝑋̄1 − 𝑋̄2 ) − 𝑡𝛼,𝜈 √ 1 + 2 ) = 1 − 𝛼
4. Pair of Observations, the Paired T-Test
𝐷̄ −𝛥0
𝑇0 = 𝑆𝐷
mD = m1 - m2
H1: 1  2  0
reject H0 if T0 > t/2,n-1 or if T0 < t/2,n-1
H1: 1  2 > 0
reject H0 if T0 > t,n-1
⁄
√𝑛
 = level of significance = type 1 error
 = type 2 error
Cheat Sheet of Doom and Despair
Power = 1-
H1: 1  2 < 0
reject H0 if T0 < t,n-1
P-Values

H1: m1 - m2 ¹ D0 à P  2P t n 1  T0

H1: m1 - m2 > D0 à P = P(tn-1 > T0)
H1: 1  2 < 0  P = P(tn-1 < T0)
Confidence Intervals
𝑆
Upper: 𝑃 (𝜇1 − 𝜇2 ≤ 𝐷̄ + 𝑡𝛼,𝑛−1 𝐷 ) = 1 − 𝛼
√𝑛
𝑆
Lower: 𝑃 (𝜇1 − 𝜇2 ≥ 𝐷̄ − 𝑡𝛼,𝑛−1 𝐷 ) = 1 − 𝛼
√𝑛
5. Ratio of Variances of 2 Normal Populations (F Distrubution)
𝑆 2 ⁄𝜎12
𝐹 = 12
n1 - 1 numerator DOF and n2 - 1 denominator DOF
𝑆2 ⁄𝜎22
reject H0 if F0  f  / 2, n1 1, n2 1 or if F0  f 1 / 2, n1 1, n2 1
𝑆2
𝐹0 = 12
𝑆2
If H1: 𝜎12 > 𝜎22
reject H0 if F0  f  , n1 1, n2 1
If H1: 𝜎12 < 𝜎22
reject H0 if F0  f 1 , n1 1, n2 1
P-Values

 
H1:  1   2 à P  2 min P f n1 1, n2 1  F0 , P f n1 1, n2 1  F0
2
2




H1:  1   2 à P  P f n1 1, n2 1  F0
2
2
H1:  1   2 à P  P f n1 1, n2 1  F0
2
2

Confidence Intervals
𝜎2
𝑆2
𝜎2
𝑆2
Upper: 𝑃 ( 12 ≤ 12 𝑓𝛼,𝑛2−1,𝑛1−1 ) = 1 − 𝛼
𝜎2
𝑆2
𝜎2
𝑆2
Lower: 𝑃 ( 12 ≥ 12 𝑓1−𝛼,𝑛2−1,𝑛1−1 ) = 1 − 𝛼
6. Two Population Proportions (p1 – p2)
𝑍0 =
𝑝̂1 −𝑝̂2
1
1
√𝑝̂(1−𝑝̂)(𝑛 +𝑛 )
1
2
𝑝̂ =
𝑋1 +𝑋2
𝑛1 +𝑛2
If H1: p1  p2
reject H0 if Z0 > Z/2 or if Z0 < Z/2
If H1: p1 > p2
reject H0 if Z0 > Z
If H1: p1 < p2
reject
P-Values
H0 if Z0 < Z
 = level of significance = type 1 error
 = type 2 error
Cheat Sheet of Doom and Despair
Power = 1-

H1: p1 ¹ p2 à P  2P Z  Z 0
 H : p > p à P = P(Z > Z )
1
1
2
H1: p1 < p2 à P = P(Z < Z0)
0
Type 2 Error/OC Function
𝑝̄ =
q1 = 1 - p1
q2 = 1 - p2
H1: p1 ¹ p2
𝛽 = 𝑃(
H1: p1 > p2
𝛽 = 𝑃 (𝑍 ≤ 𝛼
H1: p1 < p2
𝛽 = 𝑃 (𝑍 ≥
𝑛1 𝑝1 +𝑛2 𝑝2
𝑞̄ =
𝑛1 +𝑛2
−𝑍𝛼/2 √𝑝̄ 𝑞̄ (1⁄𝑛1 +1⁄𝑛2 )−(𝑝1 −𝑝2 )
≤𝑍≤
𝜎
𝑛1 𝑞1 +𝑛2 𝑞2
𝑍𝛼/2 √𝑝̄ 𝑞̄ (1⁄𝑛1 +1⁄𝑛2 )−(𝑝1 −𝑝2 )
𝜎
𝑍 √𝑝̄ 𝑞̄ (1⁄𝑛1 +1⁄𝑛2 )−(𝑝1 −𝑝2 )
)
𝜎
−𝑍𝛼 √𝑝̄ 𝑞̄ (1⁄𝑛1 +1⁄𝑛2 )−(𝑝1 −𝑝2 )
𝜎
)
Sample Size (n = n1 = n2)
2
two-sided test
𝑛=
one-sided test
𝑛=
(𝑍𝛼/2 √(𝑝1 +𝑝2 )(𝑞1 +𝑞2 )/2+𝑍𝛽 √𝑝1 𝑞1 +𝑝2 𝑞2 )
(𝑝1 −𝑝2 )2
(𝑍𝛼 √(𝑝1 +𝑝2 )(𝑞1 +𝑞2 )/2+𝑍𝛽 √𝑝1 𝑞1 +𝑝2 𝑞2 )
2
(𝑝1 −𝑝2 )2
Confidence Intervals
𝑞̂1 = 1 − 𝑝̂1
𝑞̂2 = 1 − 𝑝̂2
Upper: 𝑃(𝑝1 − 𝑝2 ≤ 𝑝̂1 − 𝑝̂2 + 𝑍𝛼 𝜎̂) = 1 − 𝛼
Lower: 𝑃(𝑝1 − 𝑝2 ≥ 𝑝̂1 − 𝑝̂2 − 𝑍𝛼 𝜎̂) = 1 − 𝛼
𝑝̂ 𝑞̂
𝑝̂ 𝑞̂
𝑛1
𝑛2
𝜎̂ = √ 1 1 + 2 2
𝑝 𝑞
𝑝 𝑞
𝑛1
𝑛2
𝜎=√ 1 1+ 2 2
𝑛1 +𝑛2
)