Stat 170B: Formula Sheet
1
• s =
n−1
2
Pn
2
i=1 xi
Pn
( i=1 xi )2
−
n
• One population mean tests:
ȳ − µ0
s
√
n
ȳ − µ0
– Test statistic: t =
s
√
n
ȳ − µ0
– Test statistic: z = σ
√
n
– Test statistic: z =
• Comparison of two population means:
ȳ1 − ȳ2 − ∆0
– Test statistic: z = s
S12
S2
+ 2
n1
n2
ȳ1 − ȳ2 − ∆0
– Test statistic: z = s
σ12
σ2
+ 2
n1
n2
– For paired data, test statistic: t =
d¯ − ∆0
√ ∼ tn−1
sD / n
√
– For paired data, the CI for µD is d¯ ± tα/2,n−1 .sD / n
– Pooled Case:
ȳ1 − ȳ2 − ∆0
∗ Test statistic: t = s ∼ tn1 +n2 −2
1
1
Sp2
+
n1
n2
Sp2 =
with
(n1 − 1)S12 + (n2 − 1)S22
n1 + n2 − 2
– Unpooled Case:
2
S12
S2
+ 2
n1
n2
ν = 2
2 2
2
S1
S2
n1
n2
+
n1 − 1
n2 − 1
ȳ1 − ȳ2 − ∆0
∗ Test statistic: t = s
∼ tν
S12
S22
+
n1
n2
with
• One population variance tests:
– Test statistic: χ20 =
– Confidence interval:
(n − 1)S 2
∼ χ2n−1
σ02
(n − 1)S 2
(n − 1)S 2
≤ σ2 ≤ 2
2
χα/2,n−1
χ1−α/2,n−1
• Comparison of two population variances:
– Test statistic: F0 = S12 /S22 when S12 > S22
– Confidence interval:
S12
σ2
S2
.F1−α/2,n2 −1,n1 −1 ≤ 12 ≤ 12 .Fα/2,n2 −1,n1 −1
2
S2
σ2
S2
• Sample size calculation:
– x-axis for the OC curve:
|δ|
2σ