Formula Sheet for Midterm 1
Math 3080-1, Spring 2007, The University of Utah
Prof: Davar Khoshnevisan
Date: January 31, 2007
Materials not covered here will not be tested
Pages: 2 (not including title page)
• Reject when T > tα,ν , where
Testing for a mean
H0 : µ = µ0 versus Ha : µ > µ0
2
S12
S22
+
m
n
ν=
2
2
(S1 /m)
(S 2 /n)2
+ 2
m−1
n−1
X̄ − µ0
X̄ − µ0
√ and/or T =
√ .
• Test statistic: Z =
σ/ n
S/ n
• Reject when Z > zα if σ known and the population is normal, or if n is large.
[round down to the nearest integer].
• Reject when T > tα,n−1 if σ unknown, but the
population is normal, or if n is large.
• P -value similar to the one-sample tests.
r
S12
S2
• CI for µ1 − µ2 : X̄ − Ȳ ± tα/2,ν
+ 2.
m
n
• P -value = 1−Φ(Z) or the corresponding formula
for a t-test, depending on which is used.
Two-sample CI for µ1 − µ2
Testing for a proportion
H0 : p = p0 versus Ha : p > p0
p̂ − p0
• Test statistic: Z = p
p0 (1 − p0 )/n
r
• If σ’s known, then X̄ − Ȳ ± zα/2
.
σ12
σ2
+ 2
m
n
• If σ’s
runknown but n, m large, then X̄ − Ȳ ±
S12
S2
+ 2.
zα/2
m
n
• Reject when Z > zα if np0 and n(1 − p0 ) are
both at least 10.
• For smaller samples, reject if p̂ > c, and find c Paired data
such that PH0 {p̂ > c} = α from a binomial table.
• Test statistic for H0 : µD = µ0 versus Ha :
D̄ − ∆0
√ .
µD > ∆0 : T =
Two-sample mean testing
SD / n
H0 : µ1 − µ2 = ∆0 versus Ha : µ1 − µ2 > ∆0
• Reject when T > tα,n−1 (normal populations, or
X̄ − Ȳ − ∆0
large samples)
.
• Test statistic: Z = r
σ12
σ22
+
• P -value similar to the one-sample tests.
m
n
SD
• CI for µD : D̄±tα/2,n−1 √ (normal populations,
• Reject with Z > zα .
n
or
large
samples).
• P -value = 1 − Φ(Z).
Two-sample proportions
• Test statistic for H0 : p1 − p2 = 0 versus Ha :
p1 − p2 > 0 is
Two-sample mean testing (large samples)
H0 : µ1 − µ2 = ∆0 versus Ha : µ1 − µ2 > ∆0
X̄ − Ȳ − ∆0
• Test statistic: Z = r
.
S12
S22
+
m
n
p̂1 − p̂2
,
1
1
p̂q̂
+
m n
Z=s
• Reject with Z > zα .
where p̂ =
• P -value = 1 − Φ(Z).
• Reject when Z > zα (n, m large).
Two-sample statistics for µ1 − µ2
(normal populations)
• Test statistic for H0 :
Ha : µ1 − µ2 > ∆0 :
X +Y
m
n
=
p̂1 +
p̂2 .
m+n
m+n
m+n
• P -value similar to the one-sample problem.
µ1 − µ2 = ∆0 versus
• CI for p1 − p2 :
r
X̄ − Ȳ − ∆0
T = r
.
S12
S22
+
m
n
p̂1 − p̂2 ± zα/2
(n, m large).
1
p̂1 q̂1
p̂2 q̂2
+
m
n
Two variances
• Test statistic for H0 : σ12 = σ22 versus Ha : σ12 >
S 2 /σ 2
σ22 : F = 12 12 .
S2 /σ2
• Reject if F > Fα,m−1,n−1 .
• P -value = the upper tail of Fn,m .
• F1−α,ν1 ,ν2 =
• CI for
1
.
Fα,ν2 ,ν1
σ12
S12
S12
:
from
to
.
σ22
Fα/2,n,m S22
F1−(α/2),n,m S22
2