ADM 2304 Study Guide - Confidence Interval

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A SUMMARY OF TESTS AND X = Variable of Interest ADM 2304
CONFIDENCE INTERVALS © T. Quon
Quantitative (Population Means) Key Assumption: Qualitative (Population Proportions)
Samples selected RANDOMLY
One Sample
H0: µ = µ0
HA: µ µ0
Calculate
n
s
x
tstat
0
µ
=
Reject H0 if
2
α
tt stat >
Confidence Interval for
µ
µµ
µ:
±n
s
tx
2
α
Two Samples
H0: µ1 - µ2 = µ0
HA: µ1 - µ2 µ0
Independent Samples
Calculate
(
)
( )
21
021
xx
stat S
xx
t
=
µ
or
(
( )
21
2
21 xx
Stxx
±
α
Unequal Population
Variances
2
2
2
1
σσ
2
2
2
1
2
1
)( 21 n
s
n
s
Sxx +=
2
α
tbased on d. f.
= * if n1 < 30
or n2 < 30
= inf if n1 30
& n2 30
& n
> 30
Equal Population
Variances
2
2
2
1
σσ
=
2
2
1
2
n
s
n
spp +=
where
(
)
(
)
2
11
21
2
22
2
11
2
+
+
=nn
snsn
sp
2
α
t based on
2
2
1
+
n
n
d. f.
Paired Samples
Calculate differences
21 xxd =
between paired
observations:
Treat as single sample
and calculate d-bar
and sd
Calculate
n
s
d
t
d
stat
0
µ
=
or
n
s
td d
2
α
±
2
α
t based on n - 1 d. f.
Assumptions:
(a) If n1 30, and n2 30, then X not too
skewed in both populations;
(b) If n1 < 30 or n2 < 30, then X
approximately normal in respective
population.
One Proportion
H0: p = p0
HA: p p0
Calculate
n
qp
pp
z
00
0
ˆ
=
or
n
qp
zp ˆˆ
ˆ
2
α
±
Two Proportions
H0: p1 - p2 = p0
HA: p1 - p2 p0
Independent
Samples
(Large Samples
Only)
Calculate
z
p p
p p p
S
=
1 2 0
1 2
^ ^
^ ^
or
(
)
( )
21 ˆˆ
2/21 ˆˆ pp
Szpp
±
α
where
( )
21 ˆˆ pp
S
2
22
1
11 ˆˆˆˆ
n
qp
n
qp +=
*if p0 = 0, this is
21
ˆˆˆˆ
n
qp
n
qp +
where
21
21
ˆnn
xx
porp +
+
=
Same Sample
(Optional)
If testing
H0: para reference
value
HA: para > reference
value,
Reject H0 if
t > tα
=
21 ˆˆ pp
S
n
pp
n
qp
n
qp 212211 ˆˆ
2
ˆˆˆˆ ++
* see Black, Business Statistics, 5th ed., p. 358.
Assumptions:
X approx. normal
if n < 30
X not too skewed
if n 30
Assumptions:
The pop’n of
differences is
normally distributed
if n < 30
Differences are not
too skewed if n 30
Assumptions:
np >= 5
and
nq >= 5
Assumptions:
5
11 pn
5
11 qn
5
22 pn
5
22 qn
Confidence Interval:
at least
(est.) -
α
t S (est.)
If testing
H0 : para reference
value
HA: para < reference
value,
Reject H0 if
t < -tα
Confidence Interval:
at most
(est.) +
α
t S (est.)
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