FORMULAS FOR EXAM 3
Rules of the expected value and the variances:
(i) For constants a, b and c and the random variables X and Y,
E(aXbYc)=aE(x)bE(Y) c
Var(aXbYc)=a2Var(x)+b2Var(Y) 2abCov(X,Y)
Var(X)=E(X2)-[E(X)]2
(ii) For constants a1 to an and the random variables X1 to Xn,
n
n
 n
 n
Var  ai X i    ai2  Var ( X i )  2   ai  a j  Cov( X i , X j )
i 1 j 1
 i 1
 i 1
i j
Random Sample:The random variables X1, X2, ….,Xn are said to form a random sample of size n if
(i) The Xi's are independent random variables.
(ii) Every Xi's has the same probability distribution.
_
If X1, X2, ….,Xn are said to form a random sample of size n with the mean  and the variance 2, the sampling distribution of x has the mean
n
 and the variance 2/n, the sampling distribution of
x
i 1
i
has the mean n and the variance n2, and so on.
Central Limit Theorem: Let X1, X2,….,Xn be a random sample from a distribution with mean  and variance 2. Then if n is sufficiently
n
_
large (n>30), x has approximately a normal distribution with mean  and variance 2/n.
x
i 1
i
has approximately a normal distribution with
mean n and variance n2. The larger the value of n, the better the approximation.
Point estimate of a parameter : single number that can be regarded as the most plausible value of . A point estimator,
estimation.
^
Unbiased estimator:

^
^

=  + error of
^
is an unbiased estimator of  if E(  )=  for every possible value of . Otherwise, it is biased and Bias = E(  )- .
Minimum Variance Unbiased Estimator (MVUE): Among all estimators of  that are unbiased, choose the one that has minimum variance.
^
The resulting

is MVUE.
^
The Invariance Principle: Let
^
^
 1 .,  2 ,...,  m be the MLE's of the parameters  1 ,  2 ,...,  m . Then the MLE of any function h(  1 ,  2 ,...,  m )
^
^
^
of these parameters is the function h(  1 .,  2 ,..., 
m
) of the MLE's
(1) Let X1,…,Xn be a random sample of normally distributed random variables with the mean  and the standard deviation .
_
x is the method of moment and the maximum likelihood estimator of 
n
_
 ( xi  x ) 2
i 1
n

(n  1) s 2
is the method of moment and the maximum likelihood estimates of 2.
n
(2) Let X1,…,Xn be a random sample of exponentially distributed random variables with parameter .
_
1 / x is the method of moment and the maximum likelihood estimator of .
(3) Let X1,…,Xn be a random sample of binomial distributed random variables with parameter p.
X/n is the method of moment and the maximum likelihood estimator of p.
(4) Let X1,…,Xn be a random sample of Poisson distributed random variables with parameter .
_
x is the method of moment and the maximum likelihood estimator of .
The Method of Moments (MME) (one unknown parameter case)
_
Calculate E(X) then set it equal to x . Solve this one equation, for the unknown parameter .
The Method of Maximum Likelihood (MLE) (one unknown parameter case)
Likelihood function is the joint pmf or pdf of X which is the function of unknown  values when x's are observed. The maximum likelihood
estimates are the  values which maximize the likelihood function. First determine the likelihood function. Then take the natural logarithm of
the likelihood function. After this, take a first derivative with respect to each unknown  and equate it to zero. Solve this one equation, one
unknown for the unknown parameter .
One-sided (One-tailed) test:
Lower tailed (Left-sided)
Upper tailed (Right-sided)
H0: population characteristics  claimed constant value H0: population characteristics  claimed constant value
Ha: population characteristics < claimed constant value Ha: population characteristics > claimed constant value
Two-sided (Two-tailed) test: H0: population characteristics = claimed constant value
Ha: population characteristics  claimed constant value
Significance level,  = P(Type I error) = P(reject H0 when it is true)
 = P(Type II error) = P(fail to reject H0 when it is false)
Power=1- = 1-P(Type II error) = P(reject H0 when it is false)
Hypothesis testing and Confidence Intervals for Population mean, 
_
0 is the claimed constant, x is the sample mean,

and
n
Characteristics
 is known, normal distribution
Test statistics
Confidence
interval
_
are the population standard deviation of x and its estimator, respectively.
n
 is unknown for a large sample (n
>40), unknown distr.
_
z
s
 is unknown for a normal population distribution
with small sample, n40 and degrees of freedom,
v=n-1
_
x  0
z
/ n
_
 _
 
 x  z / 2
, x  z / 2

n
n

x  0
s/ n
_
s _
s 
 x  z / 2
, x  z / 2

n
n

_
t
s/ n
_
s _
s 
 x  t / 2;n 1
, x  t / 2;n 1

n
n

 z    2z  
Sample size: n=   / 2  =   / 2  where the width is w=2B.
 B   w 
2
x  0
2
Decision:
(i) In each case, you reject H0 if P-value   and fail to reject H0 (accept H0) if P-value > 
if the computed test statistics is z* if the computed test statistics is t*
Lower tailed test P-value = P(z<z*)
P-value = P(t<t*)
Upper tailed test P-value = P(z>z*)
P-value = P(t>t*)
Two tailed test
P-value = 2P(z>|z*|)=2P(z<-|z*|)
P-value = 2P(t > |t*| )=2P(t <- |t*| )
(ii) Reject H0 if:
if test statistics is z if test statistics is t
Lower tailed test z  -z
t  -t;n-1
Upper tailed test z  z
t  t;n-1
Two tailed test
|z|  z/2
|t|  t/2;n-1