OPRE207 Formula Sheet for Exam II Dr. Shitao Yang

Topics Important Concepts and Formulas
Continuous Probability
Distribution
Definition
- The probability of a given interval is defined to be the area
under the graph of the probability density function f(x)
-If X is a continuous random variable, then
P(X=any specific value)=0
Normal Distribution N(
-Symmetric about the mean , “bell shape”.
-The width of the bell-shape curve is determined by 
Normal Probability Distribution Standard Normal Distribution: 0, and =1.0
Calculate Standard Normal Probability
1. P(z Excel: =normsdist(c)
2. P(z>c) =1-P(zExcel: =1-normsdist(c)
3. P(aExcel: normsdist(b)-normdist(a)
SHORT CUT: P(-c Excel: =2*normsdist(c)-1.0
Given Probability , find z-score=?
Given P(zGiven P(z>z-score)=, Excel: =normsinv(1-)
Look at the normal table for z value corresponding to the
probability or 1-.
SHORT CUT: P(-c1.0=P(z Excel: =normsinv((1+)/2)
General Normal Probability Distributions X~N(
z-Transformation
x
z




P(XGiven Probability , find x=?
Given P(XGiven P(X>x)=, Find x=? Excel: =norminv(1-)
If you transform X to z, then after you find z-score

OPRE207 Formula Sheet for Exam II Dr. Shitao Yang

Exponential Distribution  The exponential distribution has density function
where
= 0 otherwise

 ,

 Probability


 Excel Function
=EXPONDIST(


Sampling and Sampling
Distribution

1
2
1
( )
1
# " "
i n
i
i
i n
i
i
x
x
n
x x
s
n
of success out of n samples
p
n












Point Estimators
-Sample Statistics  Population Parameters
-Sample statistics are random variables!
-Population parameters are unique (constants).
Unbiased Estimators
( ) ( )x pE x E p p     
Standard Error
-Infinite Population
, ,
(1 ) (1 )
, ,
x x
p p
s
if is unknown
n n
p p p p
if p is unknown
n n

  
 
 
 
 

Central Limit Theory Large Sample Case (Approximately)
2
2
30, ~ ( , )
5 (1 ) 5, ~ ( , )
x x
p p
If n x N
If n p and n p p N
 
 

    

Precision Statement






( ) ( ) 2 ( ) 1.0
( ) ( ) 2 ( ) 1.0
x
p
P x P x P z
P p p P p p p P z

     


  

         
         
OPRE207 Formula Sheet for Exam II Dr. Shitao Yang

Interval Estimate for Population
Mean
Confidence Interval
 is known:




 is unknown:

Look at t-Table for

Sample size


Remember: Round up!
Interval Estimate for Population
Proportion

Confidence Interval:

unknownispifpUse 






Sample size:


unknownispifpUse

Remember: Round up!

n
zxIC


2
:.. 
n
s
txIC
n 1,
2
:..

 
2
22
2/ )(

zn 
n
pp
zpIC
)1(
:..
2

 
2
2
2/ )1()(

 ppzn


OPRE207 Formula Sheet for Exam II Dr. Shitao Yang

Formulate Hypotheses




Hypothesis Test

is true is false
Fail to reject No Error Type II Error
Reject Type I Error No Error

The null and alternative hypotheses should be formulated so
that the rejection of supports the conclusion and action
being sought. In such cases, the research hypothesis should be
expressed as the alternative hypothesis.

Please see the slides: Summary of Hypothesis Test (All are
important for Final Exam but Two Tailed Hypothesis Test is
particularly important for Exam II).

Excel Functions:





For a two-tailed hypothesis test, confidence interval approach,
the critical value approach, and p-value approach are equivalent
and always reach the same conclusion.

For all hypothesis tests the critical value approach and p-value
approach are equivalent and always reach the same conclusion.