MATHEMATICAL STATISTICS
(ACSC12/71-200)
Week 11 – CONFIDENCE INTERVALS
KEY IDEAS & FORMULAS
100 1 − 𝛼 % CONFIDENCE INTERVALS
IDEA: Find 𝐿(𝑋1 , … , 𝑋𝑛 ) & 𝑈 𝑋1 , … , 𝑋𝑛 such that Pr 𝐿 < 𝜃 ≤ 𝑈 = 1 − 𝛼
GENERIC FORM: Estimate ± Multiplier × SD(Estimate)
PIVOT-BASED:

𝑄=

𝑄=
𝑘𝑆
~𝐹 𝑞 ,
𝜃
𝑆−𝜃
~𝐹 𝑞
𝑘
𝑞𝛽 = 𝐹 −1 𝛽 ⟹ 𝐿 = 𝑘𝑆/𝑞1−𝛼/2 & 𝑈 = 𝑘𝑆/𝑞𝛼/2
, 𝑞𝛽 = 𝐹 −1 𝛽 ⟹ 𝐿 = 𝑆 − 𝑘𝑞1−𝛼/2 & 𝑈 = 𝑆 − 𝑘𝑞𝛼/2
CLT-BASED:

MLE: 𝜃𝑀𝐿𝐸 ± 𝑧1−𝛼/2 𝐼 −1 𝜃𝑀𝐿𝐸

SINGLE PROPORTION: 𝑝 ± 𝑧1−𝛼/2 𝑝(1 − 𝑝)/𝑛
𝑠
SINGLE AVERAGE: 𝑋 ± 𝑡1−𝛼/2 𝑛 − 1 𝑛

𝑝1 (1−𝑝1 )
𝑝2 (1−𝑝2 )
+
𝑛1
𝑛2

DIFFERENCE IN PROPORTIONS: 𝑝1 − 𝑝2 ± 𝑧1−𝛼/2

DIFFERENCE IN AVERAGES: 𝑋 − 𝑌 ± 𝑡1−𝛼/2 (𝑛 + 𝑚 − 2)𝑠𝑝
[NOTE: Pooled SD is 𝑠𝑝 =
BOOTSTRAP-t: 𝜃 − 𝑞1−𝛼/2
𝑛−1
𝑠𝑋2
+ 𝑚−1
𝑠𝑌2
/{𝑛 + 𝑚 − 2}.]
𝑉𝑎𝑟𝐵𝑂𝑂𝑇 𝜃 , 𝜃 − 𝑞𝛼/2 𝑉𝑎𝑟𝐵𝑂𝑂𝑇 𝜃
1/𝑛 + (1/𝑚)
KEY IDEAS & FORMULAS
100 1 − 𝛼 % CONFIDENCE INTERVALS
TRANSFORMATION: If (𝐿, 𝑈) a CI for 𝜃, then CR for 𝜏(𝜃) is:
{𝜏(𝑡): 𝐿 < 𝑡 < 𝑈}
UNEQUAL TAILS: Instead of 𝑞𝛼/2 and 𝑞1−𝛼/2 , use 𝑞𝛽 and 𝑞1−𝛼+𝛽 .