CHAPTER 8: CONFIDENCE INTERVAL ESTIMATION
> CHARACTERISTICS OF ESTIMATORS :
◦ unbiased ness :
expectation of the estimator equals the parameter value .
◦
Consistency : Unbiased estimator that approaches population values as sample size increases .
Efficiency Unbiased estimator that has minimal standard error
◦ : .
> A confidence interval estimate = a
range of numbers ,
called an interval ,
constructed around the point estimate
GENERAL FORMULA : A point estimate is the
value of a single sample
Point Estimate ± ( critical value ) ( standard Error ) statistic ,
such as the mean
,
CONFIDENCE INTERVAL ESTIMATE FOR THE MEAN 10 Known )
◦ ASSUMPTIONS :
population standard deviation 0 IS known
population is
normally distributed
If population is not normal ,
use large sample In 30 )
CONFIDENCE INTERVAL ESTIMATE :
-
± Za, , O : point estimate
n
✗
12
: the normal distribution critical Value for a
probability Of ✗ 12 In each tail
" " " "" " ""
EXAMPLE used
:
commonly confidence
◦
1 Consider a 95.1 .
confidence interval : £42 = I 1.96 intervals are : 90 1.
-
,
95 1
'
.
I ✗ 0.95 ✗ and 99.1
'
-
= . . = 0.05 .
'0
I >
↑
a
2=0 =
0.025
.
V
ZX / 2=-1.96 O ZX / 2=1.96
2- units :
✗ Units : Lower confidence point upper confidence
Limit Estimate Limit
, >
SAMPLING ERROR :
sampling due single sample from the population
•
is the variation
error that occurs to
selecting a .
>
The size of the sampling error =
primarily based on the amount of variation in the population and on the
sample size .
>
Large samples have less large to select
sampling error than small samples , but samples cost more .
* The value of 2- ✗ 12 needed to construct a confidence interval is called the critical value for the distribution
CONFIDENCE INTERVAL ESTIMATE FOR THE MEAN 10 unknown )
o If the population standard deviation 0 Is unknown
,
we can substitute the sample standard deviation ,
S
1 This introduces an extra uncertainty ,
since S Is a variable from sample to sample .
> distribution normal distribution
'
. . Use the t instead Of the
o ASSUMPTIONS :
population standard deviation is unknown
population is normally distributed
CONFIDENCE INTERVAL ESTIMATE i
s
t ✗ 12
-
n
>
STUDENT 'S T DISTRIBUTION NOTE : t > 2- as n increases
• The tan value depends on degrees of freedom Idf )
df = n -
l standard normal
fi)
Lumber Of Observations that are free to (t with df =
X )
vary after sample mean has been calculated
no , ,},
g.) [ tldf
(
= 5)
t
> CHARACTERISTICS OF ESTIMATORS :
◦ unbiased ness :
expectation of the estimator equals the parameter value .
◦
Consistency : Unbiased estimator that approaches population values as sample size increases .
Efficiency Unbiased estimator that has minimal standard error
◦ : .
> A confidence interval estimate = a
range of numbers ,
called an interval ,
constructed around the point estimate
GENERAL FORMULA : A point estimate is the
value of a single sample
Point Estimate ± ( critical value ) ( standard Error ) statistic ,
such as the mean
,
CONFIDENCE INTERVAL ESTIMATE FOR THE MEAN 10 Known )
◦ ASSUMPTIONS :
population standard deviation 0 IS known
population is
normally distributed
If population is not normal ,
use large sample In 30 )
CONFIDENCE INTERVAL ESTIMATE :
-
± Za, , O : point estimate
n
✗
12
: the normal distribution critical Value for a
probability Of ✗ 12 In each tail
" " " "" " ""
EXAMPLE used
:
commonly confidence
◦
1 Consider a 95.1 .
confidence interval : £42 = I 1.96 intervals are : 90 1.
-
,
95 1
'
.
I ✗ 0.95 ✗ and 99.1
'
-
= . . = 0.05 .
'0
I >
↑
a
2=0 =
0.025
.
V
ZX / 2=-1.96 O ZX / 2=1.96
2- units :
✗ Units : Lower confidence point upper confidence
Limit Estimate Limit
, >
SAMPLING ERROR :
sampling due single sample from the population
•
is the variation
error that occurs to
selecting a .
>
The size of the sampling error =
primarily based on the amount of variation in the population and on the
sample size .
>
Large samples have less large to select
sampling error than small samples , but samples cost more .
* The value of 2- ✗ 12 needed to construct a confidence interval is called the critical value for the distribution
CONFIDENCE INTERVAL ESTIMATE FOR THE MEAN 10 unknown )
o If the population standard deviation 0 Is unknown
,
we can substitute the sample standard deviation ,
S
1 This introduces an extra uncertainty ,
since S Is a variable from sample to sample .
> distribution normal distribution
'
. . Use the t instead Of the
o ASSUMPTIONS :
population standard deviation is unknown
population is normally distributed
CONFIDENCE INTERVAL ESTIMATE i
s
t ✗ 12
-
n
>
STUDENT 'S T DISTRIBUTION NOTE : t > 2- as n increases
• The tan value depends on degrees of freedom Idf )
df = n -
l standard normal
fi)
Lumber Of Observations that are free to (t with df =
X )
vary after sample mean has been calculated
no , ,},
g.) [ tldf
(
= 5)
t
