# Lecture Notes March 8, 2007. (Word doc, 54 KB.)

```Lecture Notes
AMS312 Spring2007
Mar 8th
Review of the mid-term (March 20th, in class, close book)
Ch4, Ch5 (section5.1—section5.5)
Normal binomial
Gamma  Chi  square  Exponential, passion

Moment generating function: M X (t )  E (e ) 
tX
e
tX
f X ( x) dx

 Point estimatorMLE, MOME
Properties of point estimatorsunbiased estimator (efficient estimator, best estimator,
Cramer-Rao lower bound)
 Interval Estimation:
Inference on one population mean
Inference on population proportion
Derivation of the confidence intervals
Derivation of sample size based on the maximum error E (also called the Marginal error).
Maximum error E  length of the confidence interval L
For the same  , L=2E
Problem 5.3.22
Given that a political poll shows that 52% of the sample favors candidate A, whereas 48% would
vote for candidate B, and given that the margin of error associated with the survey is 0.05, does it
make sense to claim that the two candidates are tied? Explain.
Solution: Sample size calculation based on the maximum error E for the population proportion p
Question: If we did the Harvey’s farm problem with q instead of using p, will this make a
difference of the sample size determination?
Example: Harvey’s farm birds (Turkey p; Peacock q=1-p)
1、 Let p denote the proportion of turkeys:
P(| p  pˆ | E )  1  
P( E  p  pˆ  E )  1  
P(
n
E

pˆ (1  pˆ )
n
p  pˆ

pˆ (1  pˆ )
n
E
)  1
pˆ (1  pˆ )
n
( Z )2 ˆp ( 1 ˆp )
2
E2
2、 Let q denote the proportion of peacocks
1
Lecture Notes
AMS312 Spring2007
P(| q  qˆ | E )  1  
P( E  q  qˆ  E )  1  
E
q  qˆ
E
P(


)  1
qˆ (1  qˆ )
qˆ (1  qˆ )
qˆ (1  qˆ )
n
n
n
2
2
( Z ) qˆ (1  qˆ ) ( Z ) pˆ (1  pˆ )
2
2
n

E2
E2
Hence we can see that the sample sizes we calculated are the same.
Problem 5.2.8
Solution: 1) MLE for p: pˆ 
n
1011
(we put X=6 to the group of 6+)

 X i 1536
Expected number of cars with 1 occupant=1011*P(X=1) = 1011*
2) MOME for p is the same with MLE in this problem.
Problem 5.2.14:
Solution: MLE of  when  is unknown:
2
MLE of  when  is known:
2
(X
i
 X )2
n
(X
i
n
2
  )2
1011
1011 11
(1 
)
1536
1536
```