Transcript Chapter 7 Point Estimation - University of South Alabama

Chapter 7
Point Estimation
Nutan S. Mishra
Department of Mathematics and Statistics
University of South Alabama
Estimation of population mean
When population mean µ is unknown it is estimated by the
sample mean X .
Also we have seen in the last chapter that X is an
unbiased and consistent estimator of the population
mean
That is E( X ) =µ and var( X ) 0 as n∞
Standard deviation of X = σ/√n
Estimated standard deviation of X = s /√n
Also known as estimate of standard error of sample mean
The difference between X and µ is called error in
estimation.
Point Estimation
Sample mean x assigns a single value to the unknown
value of µ . Thus it is called a point estimator.
Similarly the sample standard deviation s is a point
estimator for the unknown value of population standard
deviation σ.
The value of x is bound to be different from the
population mean µ
The difference between the two is called sampling error
Or error in estimation
Error of estimate
E = | X - µ| is the error of estimate.
To examine this error use the fact that for large n
x
~ approximate N(0,1)

n
1-α
α/2
α/2
-4
-3
-2
-1
-Zα/2
P(-Zα/2 <
x

n
< Zα/2) = 1-α
0
1
2
Zα/2
3
4
Maximum
error
of
estimate
x
P(-Zα/2 <
< Zα/2) = 1-α

n
Is equivalent to| x   | ≤ Zα/2

n
Thus the maximum error of the estimate is
E = Zα/2 .σ/√n
Thus for given value of n, σ and α we can compute the
maximum error in estimation.
Where α is the probability of error E or more.
1-α is probability that error will be smaller than E
Determining sample size
If the maximum allowable error is specified
with its probability of occurrence
E = Zα/2 .σ/√n
Then the sample size can be computed by
plugging in the other quantities in the
above equation.
This method requires prior knowledge of σ
and an assumption that n is large.
Determining sample size
If σ is unknown, and we assume that population is
approximately normal then
x
~ t-distribution with (n-1)d.f.
t
s
n
E = tα/2 .s/√n
Exercise 7.2(a)
X~ B(n, p)
The population parameter p is unknown and x/n is
an estimator of p
That is pˆ  x / n
To show that x/n is an unbiased estimator of p
That is to show that E ( p
ˆ)  p