8 Apr 2012

School

Department

Course

Professor

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.)