Formulas compiled for Statistics for economics For Mr. Iftekharul Haque's class

1. CHAPTER 7
Sampling Distribution of Sample Mean
 Population distribution
μ
X
N
i

18  20  22  24
 21 σ 
4
 (X
i
 μ)2
N
 2.236
 N= Population size n= sample size
 Sample mean X 
1 n
 Xi
n i 1
 Standard Error of mean σ X 
μX  μ
 Normal distribution

σ
(standard error of the mean decreases as the sample size increases)
n
σX 
X = sample mean μ = population mean
σ
n
σ = population standard deviation
Z
Then, Z-value for the sampling distribution of X
n = sample size
(X  μ) (X  μ)

σ
σX
n
 Finite population correction----n > 5% of N
σ2 N  n
n N 1
σX 
σ
n
Nn
N 1

Then, Var( X) 

If the n is not small compared to the N , then use Z 
 Sampling distribution property---- as n increases,
(X  μ)
σ Nn
n N 1
σ x decreases
Central Limit Theorem: As n increases Sampling Distribution becomes normal. For n>25
Central Tendency μ X  μ Variation σ X 
σ
n
 the interval - zα/2 to zα/2 encloses probability 1 – α
Then
μ  z/2σ X
is the interval that includes X with probability 1 – α

2. Sampling Distribution of Sample Proportion:
 P= population proportion

ˆ X
ˆ
P = sample proportion P 
n
ˆ
0≤ P≤1
ˆ
P has a binomial distribution, but can be approximated by a normal distribution when
nP(1 – P) > 9

ˆ
E( P)  p

Z
 X  P(1  P)
σ 2  Var   
ˆ
P
n
n
ˆ
PP

σP
ˆ
ˆ
PP
P(1  P)
n
Chapter 8
 If P(a <  < b) = 1 - 
(1 - ) is called the confidence level
then the interval from a to b is called a 100(1 - )% confidence interval of .
 Confidence Intervals for σ2 Known: z table

Point Estimate ± (Reliability Factor)(Standard Error)
x  z α/2

σ
n
Where margin of error= ME  z α/2
σ
n
W= 2ME
 The margin of error can be reduced if

the population standard deviation can be reduced (σ↓)

The sample size is increased (n↑)

The confidence level is decreased, (1 – ) ↓
 Confidence Intervals for σ2 Unknown: t table
x μ
x =mean, s=standard deviation
s/ n

t

degrees of freedom= v= n-1

P(t n 1  t n 1,α/2 )  α/2

3. 
x  t n -1,α/2
S
S
 μ  x  t n -1,α/2
n
n
where tn-1,α/2 is the critical value
 Confidence Intervals for the Population Proportion, p:
P(1  P)
n

σP 

ˆ
p  z α/2
ˆ
ˆ
ˆ
ˆ
p(1  p)
p(1  p)
ˆ
 P  p  z α/2
n
n
Chapter 10
 Key: Outcome
(Probability)
 The power of a test is the probability of rejecting a null hypothesis that is false
 Power = P(Reject H0 | H1 is true)
 Power of the test increases as the sample size increases
If Calculated z < Critical zα = Do not reject Ho
If Calculated z > Critical zα = Reject Ho

4.  p-Value Approach to Testing
 Smallest value of  for
which H0 can be rejected
 For two-tail
 Tests of the Population Proportion:
σp 
ˆ
P(1  P)
n